aa r X i v : . [ m a t h . C O ] A ug From G-parking functions to B-parking functions ∗ Fengming Dong † Mathematics and Mathematics EducationNational Institute of EducationNanyang Technological University, Singapore
Abstract
A matching M in a multigraph G = ( V, E ) is said to be uniquely restricted if M is the only perfect matching in the subgraph of G induced by V ( M ) (i.e., theset of vertices saturated by M ). For any fixed vertex x in G , there is a bijectionfrom the set of spanning trees of G to the set of uniquely restricted matchings ofsize | V | − S ( G ) − x , where S ( G ) is the bipartite graph obtained from G bysubdividing each edge in G . Thus the notion “uniquely restricted matchings of abipartite graph H saturating all vertices in a partite set X ” can be viewed as anextension of “spanning trees in a connected graph”. Motivated by this observation,we extend the notion “G-parking functions” of a connected multigraph to “B-parkingfunctions” f : X → {− , , , , · · ·} of a bipartite graph H with a bipartition ( X, Y )and find a bijection ψ from the set of uniquely restricted matchings of H to the set ofB-parking functions of H . We also show that for any uniquely restricted matching M in H with | M | = | X | , if f = ψ ( M ), then P x ∈ X f ( x ) is exactly the number of elements y ∈ Y − V ( M ) which are not externally B-active with respect to M in H , where thenew notion “externally B-active members with respect to M in H ” is an extension of“externally active edges with respect to a spanning tree in a connected multigraph”. MSC : 05A19, 05B35 and 05C85
Keywords : graph, spanning tree, parking function, bijection
The notion of a parking function was introduced by Konheim and Weiss [11] in 1966.Suppose that there are n drivers labeled 1 , , · · · , n and n parking spaces arranged ina line numbered 1 , , · · · , n . Assume that these n drivers enter the parking area in theorder 1 , , · · · , n and driver i parks at space j , where j is the minimum number with f ( i ) ≤ j ≤ n such that space j is unoccupied by the previous drivers and f ( i ) is the ∗ This paper was partially supported by NTU AcRF project (RP 3/16 DFM) of Singapore. † Email: [email protected]. i . If all drivers can park successfully by this rule, then( f (1) , f (2) , · · · , f ( n )) is called a parking function of length n . Mathematically, a function f : N n → N n , where N n = { , , · · · , n } , is called a parking function if the inequality |{ ≤ i ≤ n : f ( i ) ≤ k }| ≥ k holds for each integer k : 1 ≤ k ≤ n . For example, for n = 2,( f (1) , f (2)) = (1 , f (1) , f (2)) = (1 ,
2) and ( f (1) , f (2)) = (2 ,
1) are parking functions,but ( f (1) , f (2)) = (2 ,
2) is not. It can be shown easily that f : N n → N n is a parkingfunction if and only if there is a permutation π , π , · · · , π n of N n such that f ( π j ) ≤ j holdsfor all j = 1 , , · · · , n . Konheim and Weiss [11] proved that that the number of parkingfunctions of length n is equal to ( n + 1) n − , which is equal to the number of spanning treesof the complete graph K n +1 ([1, 3]).The parking function and its various extensions have been studied by many researchers[4, 5, 7, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30]. One of theextensions, due to Postnikov and Shapiro [18], was from parking functions to G-parkingfunctions for connected multigraphs without loops.Before talking about G-parking functions, let’s first introduce the notations of graphs usedin this article. Unless stated otherwise, we always assume that(i) G = ( V, E ) is a connected multigraph without loops, where V = { x , x , · · · , x n } and E = { y , y , · · · , y m } . For any non-empty subsets V ′ of V and E ′ of E , let G [ V ′ ]and G [ E ′ ] be the subgraphs of G induced by V ′ and E ′ respectively;(ii) H is a simple and bipartite graph with a bipartition ( X, Y ), where X = { x , x , · · · , x n } and Y = { y , y , · · · , y m } ; and(iii) H G,x is the special bipartite graph S ( G ) − x with a bipartition ( X, Y ), where x is a fixed vertex in G , S ( G ) is obtained from G by subdividing each edge in G , X = V − { x } and Y = E . An example of H G,x is shown in Figure 1.Both graphs G and H have fixed weight functions which are used for comparing edges in G or elements of Y in H . The weight function for G is an injective mapping w : E → N ,where N is the set of non-negative integers, while the weight function for H is an injectivemapping w : Y → N . Thus the weight function w of G is also the weight function of H G,x . The mapping w is injective in order to distinguish w ( y ) and w ( y ) for any distinctelements y and y .For any subsets V and V of V , let E G ( V , V ) denote the set of those edges in G joining avertex in V and a vertex in V . In particular, let E G ( u, V ) = E G ( { u } , V ) for any u ∈ V .So d G ( u ) = | E G ( u, V ) | is the degree of vertex u in G . A function f : V − { x } → N iscalled a G-parking function with respect to x if for any non-empty subset V ′ ⊆ V − { x } ,2here exists u ∈ V ′ with | E G ( u, V − V ′ ) | > f ( u ). Let GP ( G, x ) denote the set of G-parkingfunctions of G with respect to x .By Corollary 2.6, which was due to Dhar [6], a function f : V − { x } → N belongs to GP ( G, x ) if and only if there is an ordering x π , x π , · · · , x π n of vertices in V − { x } suchthat | E G ( x π i , V − V i ) | > f ( x π i ) holds for all i = 1 , , · · · , n , where V i = { x π j : i ≤ j ≤ n } .Hence a function f : N n → N n is a parking function of length n if and only if f − ∈GP ( K n +1 , V ( K n +1 ) = { , , , · · · , n } .The most interesting property on G-parking functions is the existence of bijections fromthe set of spanning trees of G , denoted by T ( G ), to GP ( G, x ). Several such bijectionshave been obtained (see [5] for example).In this paper, we focus on presenting a new extension of G-parking functions.A matching M of a graph G is said to be uniquely restricted (UR) if M is the only perfectmatching in G [ V ( M )], where V ( M ) is the set of vertices saturated by edges in M . Clearly,a matching M of G is a UR-matching if and only if | E ( C ) | > | E ( C ) ∩ M | holds for everycycle C in G , where E ( C ) is the set of edges on C . The notion of UR-matchings was firstintroduced by Golumbic, Hirst, and Lewenstein [8], originally motivated by the problemof determining a lower bound on the rank of a matrix having a specified zero/non-zeropattern. They [8] showed that the problem of finding a UR-matching with the maximumcardinality in an input graph is known to be NP-complete even for the special cases ofsplit graphs and bipartite graphs.For any T ∈ T ( G ) with E ( T ) = { y τ i : i = 1 , , · · · , n } , let M T denote the matching { x π i y τ i : i = 1 , , · · · , n } of H G,x , where x π i is the end of edge y τ i in G such that y τ i iscontained in the unique path of T connecting x and x π i . An example of T and M T isshown in Figure 1. Proposition 2.4 shows that the mapping λ defined by λ ( T ) = M T is abijection from T ( G ) to the set of UR-matchings of size n (= | V | −
1) in H G,x . ✉ ✉✉✉ ........................................................................................................................................................................................................................................................... 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(a) G (b) Graph H G,x Figure 1: E ( T ) = { y , y , y } and M T = { x y , x y , x y } The above observation shows that the notion “a spanning tree of a connected multigraph”can be viewed as a special case of the notion “a UR-matching of size | X | in a bipartitegraph H ”. Motivated by this relation, we extend the notion of G-parking functions of3onnected multigraphs to that of B-parking functions of bipartite graphs.Let U M ( H ) be the set of UR-matchings of H . For any S ⊆ X , let U M S ( H ) be the setof those members M of U M ( H ) with V ( M ) ∩ X = S . In particular, U M X ( H ) is the setof those members M of U M ( H ) with X ⊆ V ( M ). Thus U M ( H ) can be partitioned intosubsets U M S ( H ) for all subsets S of X .A mapping f : X → {− } ∪ N is called a B-parking function of H at X if for any non-empty subset S of X ( f ≥ , where X ( f ≥ = { x ∈ X : f ( x ) ≥ } , there exists x ′ ∈ S suchthat x ′ has at least f ( x ′ )+1 neighbors of degree 1 (i.e., leaves) in the subgraph of H inducedby S x ∈ S N H [ x ], where N H ( x ) is the set of neighbors of x in H and N H [ x ] = { x } ∪ N H ( x ).Let BP ( H ) be the family of B-parking functions of H at X . For any S ⊆ X , let BP S ( H )be the set of those members f ∈ BP ( H ) with X ( f ≥ = S . In particular, BP X ( H ) isthe set of those members f ∈ BP ( H ) with f ( x ) ≥ x ∈ X . Thus BP ( H ) is alsopartitioned into subsets BP S ( H ) for all subsets S ⊆ X .In Section 2, we give some basic properties on members in U M X ( H ) and members in BP X ( H ). Proposition 2.3 shows that U M X ( H ) = ∅ if and only if BP X ( H ) = ∅ . Propo-sition 2.4 shows that the members in T ( G ) correspond to members in U M X ( H G,x ) andProposition 2.5 shows that GP ( G, x ) = BP X ( H G,x ).In Section 3, we design an algorithm, called Algorithm A, for any input ( H, Y ′ ), where Y ′ ⊆ Y . Whenever U M X ( H [ X ∪ Y ′ ]) = ∅ , running this algorithm outputs a permutation π , π , · · · , π n of 1 , , · · · , n , an n -permutation τ , · · · , τ n of 1 , , · · · , m and subsets D ( x π i )of Y − Y ′ for i = 1 , , · · · , n . In this case, the mapping f : X → N defined by f ( x π i ) = | D ( x π i ) | for i = 1 , , · · · , n is a member in BP X ( H ). This result yields a mapping ψ H from U M X ( H ) to BP X ( H ). The outputs π i , τ i and D ( x π i ) for i = 1 , , · · · , n of runningAlgorithm A provide information for interpreting members in BP X ( H ).In Section 4, we show that the mapping ψ H from U M X ( H ) to BP X ( H ), defined by ψ H ( M ) = f , is a bijection, where f is the mapping from X to N defined by f ( x π i ) = | D ( x π i ) | for all i = 1 , , · · · , n , and π i and D ( x π i ) are outputs of running Algorithm A withinput ( H, V ( M ) ∩ Y ). Clearly, ψ H [ N [ S ]] provides a bijection from U M S ( H ) to BP S ( H )for every S ⊆ X , where N [ S ] = S x ∈ S N H [ x ]. Thus, there is a bijection from U M ( H ) to BP ( H ). When H is the graph H G,x , ψ H is a bijection φ G from T ( G ) to GP ( G, x ) forany connected multigraph G , where x ∈ V ( G ).In Section 5, we introduce the new notion “externally B-active members with respect to M in H ”, where M ∈ U M X ( H ), defined in Page 23, which is an extension of “externallyactive edges with respect to a spanning tree T in a connected multigraph” defined byTutte [28]. For any M ∈ U M X ( H ), if f = ψ H ( M ), then f ( x π i ) is interpreted as the4umber of those y ∈ N H ( x π i ) − (cid:0) V ( M ) ∪ S s>i N H ( x π s ) (cid:1) which are not externally B-active with respect to M in H , implying that P x i ∈ X f ( x i ) is exactly the number of thosevertices y ∈ Y − V ( M ) which are not externally B-active with respect to M in H . Thisresult implies that there exists a bijection φ G from T ( G ) to GP ( G, x ) such that for any T ∈ T ( G ), if f = φ G ( T ), then P x ∈ V ( G ) −{ x } f ( x ) is exactly the number of those edges in E ( G ) − E ( T ) which are not externally active with respect to T . In this section, we characterize UR-matchings and B-parking functions of a bipartite graph H . It is proved in Proposition 2.3 that U M X ( H ) = ∅ if and only if BP X ( H ) = ∅ . For thespecial bipartite graph H G,x , Propositions 2.4 and 2.5 show that T ( G ) and GP ( G, x )correspond to U M X ( H G,x ) and BP X ( H G,x ) respectively. By the definition of UR-matchings, the following statements are obviously equivalent forany matching M in a multigraph G :(i) M is a UR-matching of G ;(ii) M is a UR-matching of the subgraph G [ V ( M )];(iii) | E ( C ) | > | M ∩ E ( C ) | holds for any cycle C in G .For UR-matchings in a bipartite graph, another equivalent statement is given by Golumbic,Hirst and Hedetniemia [8]. Theorem 2.1 ([8]) M ∈ U M X ( H ) if and only if M = { x π i y τ i : i = 1 , , · · · , n } for apermutation π , π , · · · , π n of , , · · · , n and an n -permutation τ , τ , · · · , τ n of , , · · · , m with x π i y τ i ∈ E ( H ) for all i = 1 , , · · · , n but x π j y τ i / ∈ E ( H ) for all ≤ i < j ≤ n . Theorem 2.1 can be stated equivalently as follows.
Corollary 2.1
For any M ⊆ E ( H ) with | M | = n , M ∈ U M X ( H ) if and only if V ( M ) ∩ Y = { y τ i : i = 1 , , · · · , n } holds for some n -permutation τ , τ , · · · , τ n of , , · · · , m suchthat y τ i is a leaf in the subgraph H − S ≤ s
U M X ( H ) to be non-empty in terms of the sizes of sets N H ( S ),where S ⊆ X . Corollary 2.3
U M X ( H ) = ∅ if and only if there exists a permutation π , π , · · · , π n of , , · · · , n such that | N H ( X ) | > | N H ( X ) | > · · · > | N H ( X n ) | > , where X i = { x π j : i ≤ j ≤ n } and N H ( X i ) = S x ∈ X i N H ( x ) . By Corollary 2.3 or Theorem 2.1, if
U M X ( H ) = ∅ , then H contains at least one leaf y ′ ∈ Y in H . We are now going to show that if U M X ( H ) = ∅ , then each leaf y ′ ∈ Y of H is contained in V ( M ) for some M ∈ U M X ( H ). Proposition 2.1
Assume that y ′ ∈ Y is a leaf of H with N H ( y ′ ) = { x ′ } . Let X ′ = X − { x ′ } , H ′ = H − y ′ and H ′′ = H − { x ′ , y ′ } . The following statement hold: (i) for any M ∈ U M X ( H ) , if y ′ / ∈ V ( M ) , then M ∈ U M X ( H ′ ) ; otherwise, M −{ x ′ y ′ } ∈U M X ′ ( H ′′ ) ; (ii) if U M X ( H ) = ∅ , then y ′ ∈ V ( M ) for some M ∈ U M X ( H ) .Proof . (i) follows from Theorem 2.1 directly.6ii) Assume that M ∈ U M X ( H ) with y ′ / ∈ V ( M ). By Theorem 2.1, there exist a permu-tation π , π , · · · , π n of 1 , , · · · , n and an n -permutation τ , τ , · · · , τ n of 1 , , · · · , m suchthat M = { x π i y τ i : i = 1 , , · · · , n } and x π i y τ j / ∈ E ( H ) for all 1 ≤ j < i ≤ n .Assume that y ′ = y q and x ′ = x π k . Then τ i = q for all i = 1 , , · · · , n . Let γ k = q and γ i = τ i for all i with 1 ≤ i ≤ n and i = k . Then π , π , · · · , π n is a permutation of1 , , · · · , n and γ , γ , · · · , γ n is an n -permutation of 1 , , · · · , m such that x π i y γ i ∈ E ( H )for all i = 1 , , · · · , n but x π i y γ j / ∈ E ( H ) for all 1 ≤ j < i ≤ n . By Theorem 2.1, M ′ = { x π i y γ i , i = 1 , , · · · , n } is a member in U M X ( H ) with y ′ = y q = y γ k ∈ V ( M ′ ).Hence (ii) holds. ✷ A characterization of B-parking functions is given below.
Proposition 2.2
For any mapping f : X → N , f ∈ BP X ( H ) if and only if there is apermutation π , π , · · · , π n of , , · · · , n such that for each i = 1 , , · · · , n , x π i has at least f ( x π i ) + 1 neighbors which are leaves in the subgraph of H induced by S i ≤ j ≤ n N [ x π j ] .Proof . ( ⇒ ) Assume that f ∈ BP X ( H ). By the definition of B-parking functions, thereexists a vertex x π ∈ X such that | N H ( x π ) ∩ L ( H ) | ≥ f ( x π ) + 1.Assume that π , π , · · · , π s is a s -permutation of 1 , , · · · , n , where 1 ≤ s < n , such that forall i = 1 , , · · · , s , | N H ( x π i ) ∩ L ( H [ N [ X i ]]) | ≥ f ( x π i )+1, where X i = X −{ x π r : 1 ≤ r < i } .By the definition of B-parking functions again, there exists a vertex, denoted by x π s +1 , in X s +1 such that | N H ( x π s +1 ) ∩ L ( H [ N [ X s +1 ]]) | ≥ f ( x π s +1 ) + 1. Repeating this process, apermutation π , π , · · · , π n of N n can be obtained such that | N H ( x π i ) ∩ L ( H [ N [ X i ]]) | ≥ f ( x π i ) + 1 for all i = 1 , , · · · , n . Observe that X i is the set { x π r : i ≤ r ≤ n } . Thus thenecessity holds.( ⇐ ) Now assume that π , π , · · · , π n is a permutation of 1 , , · · · , n such that for i =1 , , · · · , n , | N H ( x π i ) ∩ L ( H [ N [ X i ]]) | ≥ f ( x π i ) + 1 holds, where X i = { x π r : i ≤ r ≤ n } .Let X ′ be an arbitrary non-empty subset of X and s be the minimum integer in N n suchthat x π s ∈ X ′ . By assumption, x π s has at least f ( x π s ) + 1 neighbors which are leaves in H [ N [ X s ]]. Observe that X ′ ⊆ X s = { x π r : s ≤ r ≤ n } , implying that for any y ∈ N H ( x π s ),if y ∈ L ( H [ N [ X s ]]), then y ∈ L ( H [ N [ X ′ ]]). Thus | N H ( x π s ) ∩ L ( H [ N [ X ′ ]]) ≥ f ( x π s ) + 1.Hence f ∈ BP X ( H ). ✷ By Proposition 2.2, one can prove the following characterization for members in BP X ( H )by acyclic orientations of H . 7 orollary 2.4 For any f : X → N , f ∈ BP X ( H ) if and only if there exists an acyclicorientation D of H such that od D ( y j ) = 1 holds for all j = 1 , , · · · , m and f ( x i ) < id D ( x i ) holds for all i = 1 , , · · · , n , where od D ( y j ) and id D ( x i ) are respectively the out-degree of y j and the in-degree of x i in D . Let f be a mapping from X to N . For any X ′ ⊆ X and x ′ ∈ X , let f | X ′ be the restrictionof f to the set X ′ and let f ( x ′ ↓ be the mapping defined by f ( x ′ ↓ ( x ′ ) = f ( x ′ ) − f ( x ′ ↓ ( x ) = f ( x ) for all x ∈ X − { x ′ } . By Proposition 2.2, we have the following result. Corollary 2.5
Assume that y ′ ∈ Y ∩ L ( H ) and N H ( y ′ ) = { x ′ } . For any mapping f from X to N , the following statements hold: (i) if f ( x ) = 0 for all x ∈ X , then BP X ( H ) = ∅ if and only if f ∈ BP X ( H ) ; (ii) f ( x ′ ↓ ∈ BP X ( H − y ′ ) if and only if f ∈ BP X ( H ) and f ( x ′ ) ≥ ; (iii) if f ( x ′ ) = 0 , then f | X −{ x ′ } ∈ BP X −{ x ′ } ( H − x ′ ) if and only if f ∈ BP X ( H ) . By applying Propositions 2.1 and 2.2, Theorem 2.1 and Corollary 2.5, it can be shownthat
U M X ( H ) = ∅ if and only if BP X ( H ) = ∅ . Proposition 2.3
The following statements are equivalent: (i) L ( H ) ∩ Y = ∅ and for each y ∈ L ( H ) ∩ Y , y ∈ V ( M ) for some M ∈ U M X ( H ) ; (ii) U M X ( H ) = ∅ ; (iii) there exist a permutation π , π , · · · , π n of , , · · · , n and an n -permutation τ , τ , · · · , τ n of , , · · · , m such that M = { x π i y τ i : i = 1 , , · · · , n } and x π i y π j / ∈ E ( H ) for all ≤ j < i ≤ n ; (iv) f ∈ BP X ( H ) , where f is the mapping defined by f ( x ) = 0 for all x ∈ X ; (v) BP X ( H ) = ∅ .Proof . Observe that (i) ⇔ (ii), (ii) ⇔ (iii), (iii) ⇔ (iv) and (iv) ⇔ (v) follow fromProposition 2.1 (ii), Theorem 2.1, Proposition 2.2 and Corollary 2.5 (i) respectively. ✷ .3 U M X ( H G,x ) and BP X ( H G,x ) We focus on the special bipartite graph H G,x in this subsection. Note that H G,x has abipartition ( X, Y ), where X = V − { x } and Y = E . Each vertex of Y is of degree 1or 2 in H G,x . As G is connected, L ( H t ) ∩ Y = ∅ for each component H t of H G,x . Alsonote that y i and y j are parallel edges in G if and only if y i and y j have the same set ofneighbors in H G,x . An example of H G,x is shown in Figure 3.In this subsection, we will show that there is a bijection from T ( G ) to U M X ( H G,x ) and GP ( G, x ) = BP X ( H G,x ) holds. Lemma 2.1 If G is a disconnected multigraph, then U M X ( H G ,x ) = ∅ .Proof . Assume that G is disconnected. Then some component of H G ,x does not haveleaves. By Corollary 2.2, U M X ( H G ,x ) = ∅ . ✷ By Lemma 2.1, we need only to consider connected multigraphs. Let T ∈ T ( G ). Withoutloss of generality, assume that E ( T ) = { y i : 1 ≤ i ≤ n } . Recall that M T denotes thematching { x ǫ i y i : i = 1 , , · · · , n } of H G,x , where x ǫ i is the end of edge y i in G such that y i is contained in the unique path in T from x to x ǫ i . By the definition of M T , M T ischaracterized by the following lemma. Lemma 2.2 M T = { x π i y τ i : i = 1 , , · · · , n } if and only if π , π , · · · , π n is a permutationof , , · · · , n such that each y τ i is an edge in E T ( V i , V − V i ) incident with x π i , where V i = { x } ∪ { x π j : 1 ≤ j < i } for i = 1 , , · · · , n . ✉ ✉✉✉ ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................... x x x x y y y y y ✉ ✉ ✉ x x x ✉ ✉ ✉ ✉ ✉ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... y y y y y ......................................................................................................................................................................................................... ......................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... (a) G (b) Graph H G,x Figure 3: Graphs G and H G,x Proposition 2.4
The mapping λ : T ( G ) → U M X ( H G,x ) defined by λ ( T ) = M T is abijection.Proof . Clearly, if T and T are distinct members in T ( G ), then M T = M T . Thus, itsuffices to prove the following statements: 9i) For any T ∈ T ( G ), M T ∈ U M X ( H G,x );(ii) For any T ∈ T ( G ), M T is the only member in U M X ( H G,x ) with V ( M T ) ∩ Y = E ( T );(iii) For any M ∈ U M X ( H G,x ), V ( M ) ∩ Y = E ( T ) holds for some T ∈ T ( G ).(i) Let T ∈ T ( G ). By Lemma 2.2, M T = { x π i y τ i : i = 1 , , · · · , n } , where π , π , · · · , π n is some permutation of 1 , , · · · , n such that y τ i is an edge in E T ( V i , V − V i ) with x π i asone end and V i = { x } ∪ { x π j : 1 ≤ j < i } for all for i = 1 , , · · · , n . Thus y τ i is a leaf in H G,x − S ≤ j
Proposition 2.5
For any mapping f : X → N , f ∈ GP ( G, x ) if and only if f ∈BP X ( H G,x ) .Proof . Consider the following statements:(i) f ∈ GP ( G, x );(ii) for any non-empty subset V ′ of X , there exists x j ∈ V ′ with | E G ( x j , V − V ′ ) | > f ( x j );(iii) for any non-empty subset V ′ of X , there exists x j ∈ V ′ such that x j has at least f ( x j )neighbors which are leaves in the subgraph of H G,x induced by S x i ∈ V ′ N H G,x ( x i );(iv) f ∈ BP X ( H G,x ).(i) ⇔ (ii) and (iii) ⇔ (iv) follow from the definitions of GP ( G, x ) and BP X ( H G,x )respectively. (ii) ⇔ (iii) follows from the fact that y ∈ E G ( x j , V − V ′ ) if and only if y is10 vertex in H G,x adjacent to x j and is also a leaf in the subgraph of H G,x induced by S x i ∈ V ′ N [ x i ]. Hence the result holds. ✷ A characterization on G-parking functions follows directly from Proposition 2.2 and Propo-sition 2.5. It was first obtained by Dhar [6].
Corollary 2.6 (Dhar [6])
For any f : V − { x } → N , f ∈ GP ( G, x ) if and only ifthere is a permutation π , π , · · · , π n of , , · · · , n such that | E G ( x π i , V − V i ) | > f ( x π i ) holds for each i = 1 , , · · · , n , where V i = { x π j : i ≤ j ≤ n } . Applying the notion of acyclic orientations of G , Corollary 2.6 can be equivalently statedas follows. Corollary 2.7
For any f : V − { x } → N , f ∈ GP ( G, x ) if and only if there exists anacyclic orientation D of G with x as its unique source such that f ( x i ) < id D ( x i ) holdsfor all i = 1 , , · · · , n , where id D ( x i ) is the in-degree of x i in D . In this section, we design an algorithm, called
Algorithm A , mainly for the purpose ofproducing a member f in BP X ( H ) for any Y ′ ⊆ Y with U M X ( H [ X ∪ Y ′ ]) = ∅ , as statedin Proposition 3.3. By this result, we are able to define a mapping ψ H from U M X ( H ) to BP X ( H ) which is shown to be a bijection in Theorem 4.1. The outputs of this algorithmare also applied in Section 5 to interpret the member f ∈ BP X ( H ) which corresponds toany given M ∈ U M X ( H ) under the mapping ψ H . The weight function w : Y → N of H is needed for running Algorithm A. In order todistinguish members in Y , we assume that w is injective and so w ( y ) = w ( y ) holds forany two different members y , y ∈ Y . The input for Algorithm A below is an order pair( H, Y ′ ), where Y ′ ⊆ Y . Algorithm A ( H, Y ′ ):A1: Input H with a bipartition ( X, Y ) and a subset Y ′ of Y ;A2: Set i := 1, I := X , D ( x ) := ∅ and F ( x ) := N H ( x ) for all x ∈ X ;113: Set L I := { y ∈ [ x ∈ I F ( x ) : y is a leaf in H I } , where H I is the subgraph of H induced by I ∪ (cid:0)S x ∈ I F ( x ) (cid:1) . If L I = ∅ , then outputthe message “the input does not yield a desired output” and stop;A4: If L I = ∅ , determine the member y ′ in L I with w ( y ′ ) < w ( y ) for all y ∈ L I − { y ′ } and the unique member x ′ ∈ N H ( y ′ );A5: If y ′ / ∈ Y ′ , then set F ( x ′ ) := F ( x ′ ) − { y ′ } , D ( x ′ ) := D ( x ′ ) ∪ { y ′ } and go back to StepA3;A6: If y ′ ∈ Y ′ , determine the unique number π i ∈ { , , · · · , n } and the unique number τ i ∈ { , , · · · , m } such that x π i = x ′ and y τ i = y ′ ;A7: Set I := I − { x ′ } . If | I | >
0, set i := i + 1 and go back to Step A3;A8: Output π i , τ i and D ( x π i ) for all i = 1 , , · · · , n and stop.Running Algorithm A has two possible outcomes. Let σ ( H, Y ′ ) = 0 if running AlgorithmA with inputs ( H, Y ′ ) stops with the message “the input does not yield a desired output”,and let σ ( H, Y ′ ) = 1 otherwise. In the case σ ( H, Y ′ ) = 1, running Algorithm A outputsnumbers π i , τ i and a subset D ( x π i ) of Y − Y ′ for i = 1 , , · · · , n , where π , π , · · · , π n isa permutation of 1 , , · · · , n and τ , τ , · · · , τ n is an n -permutation of 1 , , · · · , m . In thiscase, π i , τ i and D ( x π i ) are rigorously written as π i ( H, Y ′ ), τ i ( H, Y ′ ) and D ( H, Y ′ , x π i ). i = 1 i = 2 i = 3 i = 4 π i ( H , Y ) 4 3 2 1 τ i ( H , Y ) 5 6 1 2 D ( H , Y , x π i ) ∅ ∅ ∅ ∅ Table 1: π i ( H , Y ), τ i ( H , Y ) and D ( H , Y , x π i ), where Y = { y , y , y , y } Let’s consider some examples. Let H and H be bipartite graphs given in Figure 4with w ( y i ) = i . It is not difficult to verify that σ ( H , Y ′ ) = 0 for all subsets Y ′ of { y , y , y , y , y } . For graph H , we have σ ( H , Y ′ ) = 0 if Y ′ = { y , y , y , y } . But σ ( H , Y i ) = 1 for i = 1 ,
2, where Y = { y , y , y , y } and Y = { y , y , y , y } , and theoutputs are shown in Tables 1 and 2 respectively. σ ( H, Y ′ ) = 1 ” happen In this subsection, we shall know when the case “ σ ( H, Y ′ ) = 1” happens, and how theoutputs π i , τ i and D ( x π i ) are determined when it happens.12 = 1 i = 2 i = 3 i = 4 π i ( H , Y ) 4 3 1 2 τ i ( H , Y ) 5 6 4 3 D ( H , Y , x π i ) ∅ ∅ { y } { y } Table 2: π i ( H , Y ), τ i ( H , Y ) and D ( H , Y , x π i ), where Y = { y , y , y , y } ✉ y ✉ y ✉ y ✉ y ✉ y ✉ y ✉ x ✉ x ✉ x ✉ x 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✉ y ✉ y ✉ y ✉ y ✉ y ✉ x ✉ x ✉ x ✉ x ........................................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... 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(a) H (b) H Figure 4: Bipartite graphs H and H If L ( H ) ∩ Y = ∅ , then σ ( H, Y ′ ) = 0 clearly. If L ( H ) ∩ Y = ∅ , we have the followingobservations from Algorithm A. Lemma 3.1
Assume that L ( H ) ∩ Y = ∅ and y ′ is the member in L ( H ) ∩ Y such that w ( y ′ ) is the minimum. Let Y ′ ⊆ Y , Y ′′ = Y ′ − { y ′ } , H ′ = H − y ′ and H ′′ = H − { x ′ , y ′ } ,where x ′ is the only member in N H ( y ′ ) . The following observations follow from AlgorithmA: (i) if y ′ / ∈ Y ′ , then σ ( H, Y ′ ) = σ ( H ′ , Y ′ ) , and σ ( H, Y ′ ) = σ ( H ′′ , Y ′′ ) otherwise; (ii) if y ′ / ∈ Y ′ and σ ( H, Y ′ ) = 1 , then π i ( H, Y ′ ) = π i ( H ′ , Y ′ ) and τ i ( H, Y ′ ) = τ i ( H ′ , Y ′ ) for all i = 1 , , · · · , n , and D ( H, Y ′ , x ′ ) = D ( H ′ , Y ′ , x ′ ) ∪ { y ′ } and D ( H, Y ′ , x ) = D ( H ′ , Y ′ , x ) for all x ∈ X − { x ′ } . (iii) if y ′ ∈ Y ′ and σ ( H, Y ′ ) = 1 , then π = π ( H, Y ′ ) and τ = τ ( H, Y ′ ) such that y τ = y ′ and x π = x ′ , and π i ( H, Y ′ ) = π i − ( H ′′ , Y ′′ ) and τ i ( H, Y ′ ) = τ i − ( H ′′ , Y ′′ ) for all i = 2 , , · · · , n , and D ( H, Y ′ , x ′ ) = ∅ and D ( H, Y ′ , x ) = D ( H ′′ , Y ′′ , x ) for all x ∈ X − { x ′ } . Lemma 3.1 implies that when σ ( H, Y ′ ) = 1, the outputs π i and τ i are independent of thevertices in Y − Y ′ , but each set D ( x π i ) is a subset of Y − Y ′ . Now we are going to showthat when σ ( H, Y ′ ) = 1, the outputs of running Algorithm A can be determined by thefollowing result. Proposition 3.1
Let Y ′ ⊆ Y with σ ( H, Y ′ ) = 1 . Then π i , τ i and D ( x π i ) for i =1 , , · · · , n can be determined by the following statements: for i = 1 , , · · · , n , x π i y τ i ∈ E ( H ) and y τ i is the member in Y ′ ∩ L ( H i ) with the min-imum weight w ( y τ i ) , where H i denotes the subgraph of H induced by S i ≤ s ≤ n N [ x π s ] ; (ii) for i = 1 , , · · · , n , D ( x π i ) is the set of those y ∈ ( Y − Y ′ ) ∩ N ( x π i ) ∩ L ( H s ) suchthat w ( y ) < w ( y τ s ) holds for some s with s ≤ i .Proof . (i). By Lemma 3.1 (i), π i and τ i for i = 1 , , · · · , n are determined by runningAlgorithm A with input ( H [ X ∪ Y ′ ] , Y ′ ).It can be proved by induction on | X | . The result is obvious when | X | = 1.Now assume that | X | ≥
2. By Lemma 3.1, τ is determined by the fact that y τ is themember in Y ′ ∩ L ( H ) with the minimum weight w ( y τ ) and π is determined by thefact that x π is the only member in N H ( y τ ). By the inductive hypothesis, π i and τ i for i = 2 , · · · , n are determined by running Algorithm A with the input ( H ′ , Y ′ − { y τ } ),where H ′ = H [( X ∪ Y ′ ) − { x π , y τ } ]. Thus (i) holds.(ii) By Lemma 3.1, S ≤ i ≤ n D ( x π i ) consists of those y ∈ ( Y − Y ′ ) ∩ L ( H s ) with w ( y ) Corollary 3.1 Let Y ′ ⊆ Y with σ ( H, Y ′ ) = 1 . Then (i) { x π i y τ i : i = 1 , , · · · , n } is a member in U M X ( H ) ; (ii) x π j y τ i / ∈ E ( H ) for all j with j > i ; (iii) if y τ i , y τ j ∈ L ( H r ) , where r ≤ min { i, j } , then w ( y τ i ) < w ( y τ j ) if and only if i < j ,where H r is the subgraph of H induced by S r ≤ s ≤ n N [ x π s ] . When σ ( H, Y ′ ) = 1, let M H,Y ′ denote the subset { x π i y τ i : i = 1 , , · · · , n } of E ( H ). It willbe shown in Corollary 3.2 that for any T ∈ T ( G ), if Y ′ = E ( T ) and H is the graph H G,x ,then M H,Y ′ = M T .By Corollary 3.1 (i), M H,Y ′ ∈ U M X ( H ). Thus σ ( H, Y ′ ) = 1 implies that V ( M ) ∩ Y ⊆ Y ′ holds for some M ∈ U M X ( H ). Now we show that its converse statement also holds.14 roposition 3.2 Assume that Y ′ ⊆ Y . Then σ ( H, Y ′ ) = 1 if and only if V ( M ) ∩ Y ⊆ Y ′ holds for some M ∈ U M X ( H ) .Proof . By Corollary 3.1 (i), the necessity holds. It suffices to prove the sufficiency.When | X | = | Y | = 1, it is clear that the sufficiency holds. Assume that the sufficiencyholds when 2 ≤ | X | + | Y | < r . Now consider the case that | X | + | Y | = r and assume thatthere exists M ∈ U M X ( H ) with V ( M ) ∩ Y ⊆ Y ′ .As U M X ( H ) = ∅ , by Theorem 2.1, L ( H ) ∩ Y = ∅ . Let y ′ be the member in L ( H ) ∩ Y suchthat w ( y ′ ) is the minimum. If y ′ / ∈ Y ′ , then M ∈ U M X ( H ′ ) with V ( M ) ∩ ( Y − { y ′ } ) ⊆ Y ′ ,where H ′ = H − y ′ , and by the inductive hypothesis, σ ( H ′ , Y ′ ) = 1 holds. If y ′ ∈ Y ′ , then M − { x ′ y ′ } ∈ U M X ( H ′′ ), where H ′′ = H − { x ′ , y ′ } and x ′ is the only member in N H ( y ′ ),and by the inductive hypothesis, σ ( H ′′ , Y ′′ ) = 1 holds, where Y ′′ = Y ′ − { y ′ } . In bothcases, Lemma 3.1 implies that σ ( H, Y ′′ ) = 1.Hence the sufficiency holds. ✷ BP X ( H ) when σ ( H, Y ′ ) = 1 When σ ( H, Y ′ ) = 1, a special member of BP X ( H ) can be determined by the sets D ( H, Y ′ , x )’s. Proposition 3.3 For any Y ′ ⊆ Y with σ ( H, Y ′ ) = 1 , the function f : X → N determinedby f ( x ) = | D ( H, Y ′ , x ) | for all x ∈ X is a member in BP X ( H ) .Proof . We prove it by induction on | X | + | Y | . The result is obvious when | X | = | Y | = 1by Proposition 2.2. Assume that the result holds when 2 ≤ | X | + | Y | < r . Now considerthe case that | X | + | Y | = r .As σ ( H, Y ′ ) = 1, L ( H ) ∩ Y = ∅ . Let y ′ be the member in L ( H ) ∩ Y such that w ( y ′ ) is theminimum. Let x ′ be the only member in N H ( y ′ ).First consider the case that y ′ / ∈ Y ′ . By the inductive hypothesis, the function g : X → N defined by g ( x ) = | D ( H ′ , Y ′ , x ) | for all x ∈ X is a member in BP X ( H ′ ), where H ′ = H − y ′ .By Corollary 2.5(ii), the function f : X → N defined by f ( x ′ ) = g ( x ′ )+ 1 and f ( x ) = g ( x )for all x ∈ X − { x ′ } is a member in BP X ( H ). By Lemma 3.1(i), f ( x ) = | D ( H, Y ′ , x ) | forall x ∈ X . Thus the result holds in this case.Now consider the case that y ′ ∈ Y ′ . Then σ ( H ′′ , Y ′′ ) = σ ( H, Y ′ ) = 1 by Lemma 3.1 (ii),where Y ′′ = Y ′ − { y ′ } and H ′′ = H − { x ′ , y ′ } . By the inductive hypothesis, the function g : X − { x ′ } → N defined by g ( x ) = | D ( H ′′ , Y ′′ , x ) | for all x ∈ X − { x ′ } is a member15n BP X ′ ( H ′′ ), where X ′ = X − { x ′ } . By Corollary 2.5(iii), the function f : X → N defined by f ( x ′ ) = 0 and f ( x ) = g ( x ) for all x ∈ X − { x ′ } is a member in BP X ( H ). ByLemma 3.1(ii), f ( x ) = | D ( H, Y ′ , x ) | for all x ∈ X . Thus the result also holds in this case.Hence the result holds. ✷ H G,x In the next two subsections, we will consider the special bipartite graph H G,x .Note that the weight function w : E → N for edges of G is also the weight func-tion for members of Y in H G,x , which is used in running Algorithm A with input( H G,x , Y ′ ), where Y ′ ⊆ Y = E . If σ ( H G,x , Y ′ ) = 1, simply write π i = π i ( H G,x , Y ′ ), τ i = τ i ( H G,x , Y ′ ) and D ( x π ) = D ( H G,x , Y ′ , x π ) for i = 1 , , · · · , n .The next result follows from Propositions 3.1 and 3.2. Proposition 3.4 σ ( H G,x , Y ′ ) = 1 if and only if G [ Y ′ ] is a connected and spanning sub-graph of G . Furthermore, if σ ( H G,x , Y ′ ) = 1 , then, for i = 1 , , · · · , n , (i) y τ i is the edge in Y ′ ∩ E G ( V i , V − V i ) with w ( y τ i ) ≤ w ( y ′ ) for all y ′ ∈ Y ′ ∩ E G ( V i , V − V i ) and x π i is the vertex in V − V i incident with y τ i , where V i = { x } ∪ { x π s : 1 ≤ s < i } ; (ii) D ( x π i ) is the set of those edges y ∈ Y − Y ′ incident with x π i such that y ∈ E G ( V s , V − V s ) and w ( y ) < w ( y τ s ) hold for some s ≤ i . By Lemma 2.2 and Proposition 3.4 (i), for any T ∈ T ( G ), we have the following relationon M T and M H G,x ,Y ′ , where Y ′ = E ( T ). Corollary 3.2 For any T ∈ T ( G ) , M T = M H G,x ,Y ′ = { x π i y τ i : i = 1 , , · · · , n } , where Y ′ = E ( T ) . For example, let G = ( V, E ) be the graph shown in Figure 5 (a) and Y ′ be a subset of E with G [ Y ′ ] shown in Figure 5 (b), where each number beside an edge e is its weight w ( e ).As G [ Y ′ ] is a spanning tree of G , Proposition 3.4 implies that σ ( H G,x , Y ′ ) = 1.By Proposition 3.4 (i), y τ , · · · , y τ are the following edges respectively: x x , x x , x x , x x , x x , x x , x ✉ x ✉ x ✉ x ✉ x ✉ x ✉ x .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .......................................................................................................................................................................................................................................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................... 145 9 11103612 827 ✉ x ✉ x ✉ x ✉ x ✉ x ✉ x ✉ x ....................................................................................................................................................................................................... .......................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................... 14 912 87 (a) G (b) G [ Y ′ ]Figure 5: G and a spanning tree G [ Y ′ ]and x π , · · · , x π are the vertices x , x , x , x , x , x respectively. By applying Proposi-tion 3.4 (ii), we have D ( x ) = D ( x ) = D ( x ) = D ( x ) = ∅ , D ( x ) = { x x } , D ( x ) = { x x , x x , x x } . The next result considers the special case that Y ′ = E ( T ) for a given T ∈ T ( G ). It willbe applied for proving Theorem 5.1.Let P i,j denote the unique path in T connecting vertices x π i and x π j . Proposition 3.5 Let T ∈ T ( G ) and Y ′ = E ( T ) . Then (i) for i = 1 , , · · · , n , G [ E i ] is a tree with vertex set { x π s : 0 ≤ s ≤ i } , where E i = { y τ s :1 ≤ s ≤ i } and π = 0 ; (ii) for i = 1 , , · · · , n , y τ i is incident with x π i and is an edge on the path P ,i ; (iii) if x π i is a vertex on the path P ,j , then i ≤ j holds; (iv) for any integers ≤ i, j ≤ n , if max { b ( y τ i ) , b ( y τ j ) } < min { i, j } , then w ( y τ i ) < w ( y τ j ) if and only if i < j , where b ( y τ j ) is the number s such that x π s is the end of y τ j in G different from x π j .Proof . (i) follows from Proposition 3.4 (i).(ii) and (iii) follow directly from result (i).(iv). Let r = max { b ( y τ i ) , b ( y τ j ) } . As r < min { i, j } , both y τ i and y τ j are members in theset Y ′ ∩ E G ( V k , V − V k ) for all k with r < k ≤ min { i, j } , where V k = { x π t : k ≤ t ≤ n } .By Proposition 3.4 (i), w ( y τ i ) < w ( y τ j ) if and only if y τ i is selected before y τ j , i.e., i < j .Thus (iv) holds. ✷ .5 The minimum spanning tree The minimum spanning tree of G with respect to w is the spanning tree T of G suchthat w ( T ) < w ( T ) holds for all T ∈ T ( G ) − { T } , where w ( T ) = P e ∈ E ( T ) w ( e ). In thissubsection, we show that the minimum spanning tree of G is determined by the outputs y τ i ’s of running Algorithm A with input ( H G,x , E ( G )). But this property cannot beextended to all bipartite graphs.Prim’s algorithm [19] is a well-known algorithm of determining the minimum spanningtree of a connected multigraph. The way of choosing edges of the minimum spanning treein G by Prim’s algorithm (see [2, 29]) is actually the same as the way of determining edges y τ , · · · , y τ n by Proposition 3.4 (i). Thus the next result follows from Proposition 3.4 (i)and Prim’s algorithm. Corollary 3.3 For any Y ′ ⊆ E , if G [ Y ′ ] is connected and spanning, then E ( T ) = { y τ , · · · , y τ n } for the minimum spanning tree T of G [ Y ′ ] . For any Y ′′ ⊆ E with Y ′ ⊂ Y ′′ , when do G [ Y ′ ] and G [ Y ′′ ] have the same the minimumspanning tree? Theorem 3.1 Let T be the minimum spanning tree of G [ Y ′ ] . For any Y ′′ ⊆ E with Y ′ ⊆ Y ′′ , T is the minimum spanning tree of G [ Y ′′ ] if and only if (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ .Proof . It suffices to show that the two statements below hold:(a) if (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ , then T is the minimum spanning tree of G [ Y ′′ ];(b) if (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ , then T is not the minimum spanning tree of G [ Y ′′ ].Assume that (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ . By Proposition 3.4 (ii), S ≤ i ≤ n D ( x π i ) is theset of those edges y ∈ Y − Y ′ such that y ∈ E G ( V s , V − V s ) and w ( y ) ≤ w ( y τ s ) hold forsome s with 1 ≤ s ≤ n , where V s = { x π t : s ≤ t ≤ n } . As (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ ,by Proposition 3.4 (i), y τ i is the edge in E G [ Y ′′ ] ( V i , V − V i ) such that w ( y τ i ) < w ( y ) holdsfor all edges y ∈ E G [ Y ′′ ] ( V i , V − V i ) − { y τ i } for each i = 1 , , · · · , n . By Prim’s algorithm, E ( T ) = { y τ i : i = 1 , , · · · , n } is the edge set of the minimum spanning tree of G [ Y ′′ ].Hence (a) holds.Now consider the case that (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ = ∅ . By Corollary 3.3, the edge setof the minimum spanning tree T of G [ Y ′ ] is { y τ , y τ , · · · , y τ n } . By Prim’s algorithm,the edges of T can be chosen in the order y τ , y τ , · · · , y τ n . By the assumption, there18xists y ∈ (cid:16)S ≤ i ≤ n D ( x π i ) (cid:17) ∩ Y ′′ . By Proposition 3.4 (ii), y ∈ E G [ Y ′′ ] ( V s , V − V s ) and w ( y ) < w ( y τ s ) hold for some s with 1 ≤ s ≤ n . By Prim’s algorithm again, y ischosen as an edge of the minimum spanning tree of G [ Y ′′ ] at the step after all edges in { y τ t : 1 ≤ t < s } are selected, implying that T is not the minimum spanning tree of G [ Y ′′ ].Hence (b) also holds. ✷ For any M ∈ U M X ( H ), let w ( M ) = P y ∈ V ( M ) ∩ Y w ( y ). A member M in U M X ( H ) iscalled a minimum member in U M X ( H ) if w ( M ) ≤ w ( M ) holds for all M ∈ U M X ( H ). ByCorollary 3.3, { x π i y τ i : i = 1 , , · · · , n } is the unique minimum member of U M X ( H G,x ).However, this result does not hold all bipartite graphs H . An example is shown in Figure 6.Let H be the bipartite graph shown in Figure 6, where any vertex with an order pair( y i , w i ) beside is vertex y i with w ( y i ) = w i . Running Algorithm A with input ( H , Y ),where Y = { y , y , y , y } , outputs π i = τ i = i for i = 1 , , 3. But M = { x i y i : i = 1 , , } is not the minimum member of U M X ( H ), as M = { x y , x y , x y } ∈ U M X ( H ) and w ( M ) = w ( y ) + w ( y ) + w ( y ) < w ( y ) + w ( y ) + w ( y ) = w ( M ) . ✉ x ✉ x ✉ x ✉ ( y , ✉ ( y , ✉ ( y , ✉ ( y , ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ......................................................................................................................................................................................................................................................... Figure 6: A bipartite graph H Problem 3.1 For any bipartite graph H with a bipartition ( X, Y ) and U M X ( H ) = ∅ ,determine the minimum member of U M X ( H ) . ψ H from U M X ( H ) to BP X ( H ) For any M ∈ U M X ( H ), let ψ H ( M ) = f , where f is the mapping f : X → N definedby f ( x i ) = | D ( H, Y ∩ V ( M ) , x i ) | for each x i ∈ X . By Propositions 3.2 and 3.3, ψ H is amapping from U M X ( H ) to BP X ( H ). By its definition, an interpretation of ψ H is givenby Proposition 3.1. We are now going to show that ψ H is a bijection. Theorem 4.1 The mapping ψ H : U M X ( H ) → BP X ( H ) defined above is a bijection from U M X ( H ) to BP X ( H ) . roof . We first prove that ψ H is injective by induction on | X | + | Y | . When | X | = | Y | = 1,the conclusion is obvious, as U M X ( H ) has at most one member. Assume that it holdswhen | X | + | Y | < k , where k ≥ 3. Now consider the case that | X | + | Y | = k .Assume that U M X ( H ) = ∅ . By Theorem 2.1, Y ∩ L ( H ) = ∅ . Assume that y ′ is themember in Y ∩ L ( H ) such that w ( y ′ ) is the minimum. Let x ′ be the only member in N H ( y ′ ).Let M and M be distinct members in U M X ( H ) and Y i = V ( M i ) ∩ Y for i = 1 , Y = Y , then V ( M ) = V ( M ), implying that M = M by the definition of UR-matchings. Thus Y = Y . Let f i ( x ) = | D ( H, Y i , x ) | for i = 1 , x ∈ X . We shallshow that f = f in the three cases below. Case 1 : y ′ ∈ Y − Y or y ′ ∈ Y − Y .Assume that y ′ ∈ Y − Y . By Lemma 3.1, D ( H, Y , x ′ ) = ∅ while y ′ ∈ D ( H, Y , x ′ ). Thus f ( x ′ ) < f ( x ′ ). Case 2 : y ′ / ∈ Y ∪ Y .In this case, M i ∈ U M X ( H ′ ) for i = 1 , 2, where H ′ = H − y ′ . By the inductivehypothesis, ψ H ′ is an injective mapping from U M X ( H ′ ) to BP X ( H ′ ), implying that | D ( H ′ , Y , x ) | 6 = | D ( H ′ , Y , x ) | for some x ∈ X . By Lemma 3.1(i), for each i = 1 , D ( H, Y i , x ′ ) = D ( H ′ , Y i , x ′ ) ∪ { y ′ } and D ( H, Y i , x ) = D ( H ′ , Y i , x ) for all x ∈ X − { x ′ } ,implying that | D ( H, Y , x ) | 6 = | D ( H, Y , x ) | for some x ∈ X , i.e., f = f . Case 3 : y ′ ∈ Y ∩ Y .By Lemma 3.1(ii), for i = 1 , D ( H, Y i , x ′ ) = ∅ and D ( H, Y i , x ) = D ( H ′′ , Y ′ i , x ) forall x ∈ X ′ = X − { x ′ } , where H ′′ = H − { x ′ , y ′ } and Y ′ i = Y i − { y ′ } . Note that Y ′ i = Y ∩ V ( M ′ i ) for i = 1 , 2, where M ′ i = M i − { x ′ y ′ } . As M = M , we have M ′ = M ′ .By the inductive hypothesis, | D ( H ′′ , Y ′ , x ) | 6 = | D ( H ′′ , Y ′ , x ) | for some x ∈ X ′ , implyingthat | D ( H, Y , x ) | 6 = | D ( H, Y , x ) | . Thus, f = f in this case.Therefore ψ H is injective.It remains to prove that ψ H is surjective, i.e., the following statement “for any f ∈BP X ( H ), there exists M ∈ U M X ( H ) with ψ H ( M ) = f ” holds. We prove this statementby induction on the value of | X | + | Y | + P x ∈ X f ( x ), where f ∈ BP X ( H ). Observe that | X | + | Y | + P x ∈ X f ( x ) ≥ 2. When | X | + | Y | + P x ∈ X f ( x ) = 2, we have | X | = | Y | = 1and f ( x ) = 0 for the only member x ∈ X , implying that H ∼ = K and ψ H ( M ) = f holds,where M = E ( H ).Assume that the above statement holds for any bipartite graph H ′ with a bipartition20 X ′ , Y ′ ) and any f ′ ∈ BP X ′ ( H ′ ) such that | X ′ | + | Y ′ | + P x ∈ X ′ f ′ ( x ) < r , where r ≥ H is a bipartite graph with a bipartition ( X, Y ) and f ∈ BP X ( H )such that | X | + | Y | + P x ∈ X f ( x ) = r .As BP X ( H ) = ∅ , by Proposition 2.3, we have U M X ( H ) = ∅ and Y ∩ L ( H ) = ∅ . Assumethat y ′ is the member in Y ∩ L ( H ) such that w ( y ′ ) is the minimum and x ′ is the onlymember in N H ( y ′ ). We shall prove in the two cases below that ψ H ( M ) = f holds for some M ∈ U M X ( H ). Case 1’ : f ( x ′ ) = 0.Let H ′′ = H − { x ′ , y ′ } and g = f | X ′ , where X ′ = X − { x ′ } . By Corollary 2.5(iii), g ∈ BP X ′ ( H ′′ ). By the inductive hypothesis, there exists M ′ ∈ U M X ′ ( H ′′ ) such that ψ H ′′ ( M ′ ) = g , i.e., g ( x ) = | D ( H ′′ , V ( M ′ ) ∩ Y, x ) | for all x ∈ X ′ . It is clear that M = M ′ ∪{ x ′ y ′ } ∈ U M X ( H ). By Lemma 3.1(ii), D ( H, Y ′ , x ′ ) = ∅ and D ( H, Y ′ , x ) = D ( H ′′ , Y ′′ , x )for all x ∈ X − { x ′ } , where Y ′′ = V ( M ′ ) ∩ Y and Y ′ = Y ′′ ∪ { y ′ } = V ( M ) ∩ Y . Thus f ( x ′ ) = 0 = | D ( H, Y ′ , x ′ ) | and f ( x ) = g ( x ) = | D ( H ′′ , Y ′′ , x ) | = | D ( H, Y ′ , x ) | for all x ∈ X − { x ′ } , implying that ψ H ( M ) = f . Case 2’ : f ( x ′ ) > H ′ = H − { y ′ } and g = f ( x ′ ↓ . By Corollary 2.5(ii), g ∈ BP X ( H ′ ). By the inductivehypothesis, there exists M ∈ U M X ( H ′ ) such that ψ H ′ ( M ) = g , i.e., g ( x ) = | D ( H ′ , Y ′ , x ) | for all x ∈ X , where Y ′ = V ( M ) ∩ Y . By Lemma 3.1(i), D ( H, Y ′ , x ′ ) = D ( H ′ , Y ′ , x ′ ) ∪{ y ′ } and D ( H, Y ′ , x ) = D ( H ′ , Y ′ , x ) for all x ∈ X − { x ′ } . Thus f ( x ′ ) = g ( x ′ ) + 1 = | D ( H ′ , Y ′ , x ′ ) | + 1 = | D ( H, Y ′ , x ′ ) | and f ( x ) = g ( x ) = | D ( H ′ , Y ′ , x ) | = | D ( H, Y ′ , x ) | forall x ∈ X − { x ′ } , implying that ψ H ( M ) = f . ✷ For any T ∈ T ( G ), define φ G ( T ) = ψ H G,x ( M T ). By Theorem 4.1, Corollary 3.2 andProposition 2.5, φ G is a bijection from T ( G ) to GP ( G, x ). By Proposition 3.4, φ G canbe interpreted by the following result, which first appeared in [5]. Corollary 4.1 Let T ∈ T ( G ) . Assume that vertices x π , x π , · · · , x π n and egdes y τ , y τ , · · · , y τ n of G are determined by Proposition 3.4 (i), where Y ′ = E ( T ) . If f = φ G ( T ) , then, for i = 1 , , · · · , n , f ( x π i ) is the number of those edges y ′ ∈ E ( G ) − E ( T ) incident with x π i and some x π j , where ≤ j < i , with w ( y ′ ) < max j Theorem 4.1 shows that the mapping ψ H : U M X ( H ) → BP X ( H ) defined by ψ H ( M ) = f is a bijection, where f ( x ) = | D ( H, V ( M ) ∩ Y, x ) | for all x ∈ X . In this section, assumethat M ∈ U M X ( H ) and Y ′ = V ( M ) ∩ Y , unless otherwise stated. Also assume that π i = π i ( H, Y ′ ), τ i = τ i ( H, Y ′ ) and D ( x π i ) = D ( H, Y ′ , x π i ). In this section, we will give aninterpretation for f different from Proposition 3.1 (ii).In Subsection 5.1, we define a unique path P ( H,M ) ( y ) in H for each y ∈ Y − Y ′ withrespect to M . In Subsection 5.2, we introduce the concept “externally B-active memberswith respect to M in H ” by comparing w ( y ) with w ( y ′ ) for all those y ′ ∈ Y which arein the path P ( H,M ) ( y ). In Subsection 5.3, we show that S x ∈ X D ( H, Y ′ , x ) is exactly theset of those members in Y − Y ′ which are not externally B-active with respect to M in H . In particular, D ( H, Y ′ , x π i ) is the set of those members y in (( Y − Y ′ ) ∩ N H ( x π i )) − S s>i N H ( x π s ) which are not externally B-active with respect to M in H , where Y ′ = V ( M ) ∩ Y . Finally, in Subsection 5.4, we introduce a generating function Ω( H ; x, y, z ) forthe members in U M ( H ) with three variables. Particularly, Ω( H G,x ; x, y, 0) is the Tuttepolynomial T G ( x, y ). P ( H,M ) ( y ) for each y ∈ Y − Y ′ By the definition of π i and τ i for i = 1 , , · · · , n , we have Y ′ = { y τ i : i = 1 , , · · · , n } and M = M H,Y ′ = { x π i y τ i : i = 1 , , · · · , n } . For any vertex y ∈ Y and any integer j ≥ 1, let n j ( y ) = 0 if j > d H ( y ), and let n j ( y ) be the j ’th largest integer s such that x π s ∈ N ( y )otherwise. In other words, n ≥ n ( y ) > n ( y ) > · · · > n d H ( y ) ( y ) > n j ( y ) = 0 for all j > d H ( y ) and N ( y ) = { x π s : s ∈ { n ( y ) , · · · , n d H ( y ) ( y ) }} .Clearly n ( y τ i ) = i for all i = 1 , , · · · , n by Corollary 3.1 (i) and (ii). By Proposition 3.1, D ( H, Y ′ , x π i ) ⊆ { y : Y − Y ′ , n ( y ) = i } .For any y ∈ Y − Y ′ , let P ( H,M ) ( y ) be the following maximal M -alternating path in H with y as one end: P ( H,M ) ( y ) : yx π j y τ j · · · x π jt y τ jt where j = n ( y ), j i = n ( y τ ji − ) > i = 2 , , · · · , t and n ( y ) < j t , as shown inFigure 7. Thus j > j > · · · > j t > n ( y ). By the maximality of P ( H,M ) ( y ), n ( y ) ≥ n ( y τ jt ) ≥ 0. Clearly that the path P ( H,M ) ( y ) is unique for each y .For example, if H is the bipartite graph shown in Figure 8 with w ( y i ) = i for all i and M = { x i y i : i = 1 , , · · · , } ∈ U M X ( H ), then Y ′ = { y i : i = 1 , , · · · , } , π i = τ i = i for i = 1 , , · · · , 5. Note that y is the only vertex in Y − Y ′ . As n ( y ) = 5, n ( y ) = 4,22 ✉ ✉ ✉✉ ✉ ✉ ✉ ✉ .................................................................................................................................................................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................................................................................................................................................. 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.............................................................................................................................................................................................................................................................................................................................................................................................. ................................................................................................... · · · x π j x π j x π jt y τ j y τ j y τ jt y Figure 7: P ( H,M ) ( y ) : yx π j y τ j · · · x π jt y τ jt n ( y ) = 3 and n ( y ) = 2 = n ( y ), the path P ( H,M ) ( y ) is P ( H,M ) ( y ) : y x y x y x y . ✉ y ✉ y ✉ y ✉ y ✉ y ✉ y ✉ x ✉ x ✉ x ✉ x ✉ x 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Figure 8: P ( H,M ) ( y ) : y x y x y x y and n ( y ) = n ( y ) = 2 M For any y ∈ Y − Y ′ , y is the only vertex in the path P ( H,M ) ( y ) belonging to Y − Y ′ .We say y is externally B-active with respect to M in H if w ( y ) > w ( y τ jr ) holds for all r = 1 , , · · · , t , where { y τ jr : r = 1 , , · · · , t } = Y ′ ∩ V ( P ( H,M ) ( y )). Let A ex ( H, M ) denotethe set of those members in Y − Y ′ which are externally B-active with respect to M in H , and let N A ex ( H, M ) = ( Y − Y ′ ) − A ex ( H, M ). Thus N A ex ( H, M ) is the set of thosemembers in Y − Y ′ which are not externally B-active with respect to M in H .Recall that the weight function w of G is a fixed injective mapping from E to N . In-troduced by Tutte [28], for a given T ∈ T ( G ), an edge y in E ( G ) − E ( T ) is said to be externally active with respect to T if w ( y ) ≥ w ( y ′ ) holds for all edges y ′ in the unique cycleof the subgraph G [ E ( T ) ∪ { y } ], and an edge y ∈ E ( T ) is said to be internally active withrespect to T if w ( y ) ≥ w ( y ′ ) holds for every edge y ′ ∈ E ( G ) − E ( T ) with the propertythat ( E ( T ) − { y } ) ∪ { y ′ } = E ( T ′ ) holds for some T ′ ∈ T ( G ). For the definition of thesetwo concepts, the condition “ w ( y ) ≥ w ( y ′ )” can be replaced by “ w ( y ) ≤ w ( y ′ )”, as thecondition is changed when w ( e ) is replaced by K − w ( e ) for each edge e in G , where K is a number in N such that K − w ( e ) ≥ e ∈ E . Tutte [28] expressed the Tuttepolynomial T G ( x, y ) as the summation of x ia ( T ) y ea ( T ) over all spanning trees T of G , where ea ( T ) and ia ( T ) are respectively the number of externally active edges and the number ofinternally active edges with respect to T . 23n the following, we prove that the concept “externally active with respect to T ” is ex-tended to the one “externally B-active with respect to M ”, where M ∈ U M X ( H ). Theorem 5.1 Let T ∈ T ( G ) . For any y ∈ E ( G ) − E ( T ) , y is externally active respect to T in G if and only if y ∈ A ex ( H G,x , M T ) .Proof . Let Y ′ = E ( T ) and let H simply denote H G,x in the proof. Thus σ ( H, Y ′ ) = 1,and π i ’s and τ i ’s are determined by Proposition 3.4(i) and have the properties in Propo-sition 3.5.Write x π i (cid:22) x π j if x π i is a vertex on the path P ,j and x π i x π j otherwise. By Proposi-tion 3.5 (iii), Claim 1 follows directly. Claim 1 : x π i (cid:22) x π j implies that i ≤ j .Thus x π i (cid:22) x π j if and only if i ≤ j and P i,j is part of P ,j . In the following, we firstcompare i and j in the case that x π i x π j and x π j x π i . Define w max ( P i,j ) as follows: w max ( P i,j ) = (cid:26) − , if E ( P i,j ) = ∅ max { w ( e ) : e ∈ E ( P i,j ) } , otherwise . ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. x · · · x π r · · · x π s x π j y τ j ✉ x π i ✉ x π q ✉ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... · ·· Figure 9: x π r (cid:22) x π i , x π r (cid:22) x π j but E ( P r,i ) ∩ E ( P r,j ) = ∅ . Claim 2 : If x π r (cid:22) x π i , x π r (cid:22) x π j and E ( P r,i ) ∩ E ( P r,j ) = ∅ , then w max ( P r,i ) < w max ( P r,j )implies that i < j .Assume that w max ( P r,i ) < w max ( P r,j ). We shall prove Claim 2 by induction on the thevalue of ρ ( i, j ) = | E ( P r,i ) | + | E ( P r,j ) | . By Proposition 3.5 (iv) and the definition of w max ( P i,j ), Claim 2 holds when | E ( P r,i ) | ≤ | E ( P r,j ) | ≤ ρ ( i, j ) < K , where K ≥ 3. Now consider the case that ρ ( i, j ) = K .Let k be the least possible integer such that y τ k is an edge on the path P r,j with w ( y τ k ) >w max ( P r,i ). As w max ( P r,i ) < w max ( P r,j ), such k exists. By Claim 1, r < k ≤ j . If k < j ,then ρ ( i, k ) < K and by the inductive hypothesis, w max ( P r,i ) < w ( y τ k ) = w max ( P r,k )implies that i < k , and so i < j holds. Thus it suffices to consider the case that k = j ,24.e., w max ( P r,i ) < w ( y τ j ), but w max ( P r,i ) > w ( y τ t ) for all edges y τ t on the path P r,j with t = j .Let s = b ( y τ j ) and q = b ( y τ i ), as shown in Figure 9, where b ( y τ j ) is defined in Propo-sition 3.5(iv) (i.e., b ( y τ j ) is the number s such that x π s is the end of y τ j in G differentfrom x π j ). By Claim 1, q < i and s < j . As ρ ( q, j ) < K , by the inductive hypothesis, w ( y τ j ) > w max ( P r,i ) ≥ w max ( P r,q ) implies that j > q . As w max ( P r,i ) > w max ( P r,s ), wehave i > s by the inductive hypothesis. Since b ( y τ j ) = s < i and b ( y τ i ) = q < j , theinequality w ( y τ j ) > w max ( P r,i ) ≥ w ( y τ i ) implies that j > i by Proposition 3.5 (iv).Hence Claim 2 holds.Now let y be any edge in E ( G ) − E ( T ). Assume that x π i and x π j are the two ends of y ,where j > i , and the unique cycle C in the graph obtained from T by adding y consists ofedge y and two edge-disjoint paths P r,i and P r,j , where x π r (cid:22) x π i and x π r (cid:22) x π j . Thus r ≤ i < j with the possibility that i = r .Let x π j x π j · · · x π jt be the longest possible subpath of P r,j between x π j and x π jt suchthat i < j t , as shown in Figure 10. By Claim 1, we have j > j > · · · > j t > i ≥ b ( y τ jt ) , (5.1)where i = b ( y τ jt ) if and only if i = r and b ( y τ jt ) = r . ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. x · · · x π r · · · · · · x π jt x π j x π j y ✉ x π i ✉✉ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... · ·· .............................................................................................................................................................................................................................................................................. Figure 10: b ( y τ jt ) ≤ i < j t < · · · < j < j As j > i ≥ b ( y τ jt ), by Claim 2, w max ( P r,k ) ≤ w max ( P r,i ) < w max ( P r,j t ), where k = b ( y τ jt ),implying thatmax { w max ( P r,i ) , w max ( P r,j ) } = max { w ( y τ js ) : s = 1 , , · · · , t } . Thus the following claim holds. Claim 3 : y is externally active with respect to T in G if and only if w ( y ) > w ( y τ js ) holdsfor all s = 1 , , · · · , t .On the other hand, by (5.1) and the fact that n ( y ) = j , n ( y τ js ) = b ( y τ js ) = j s +1 for s = 1 , , · · · , t − n ( y ) = b ( y ) = i ≥ k = b ( y τ jt ) = n ( y τ jt ), the path P M T ( y ) in H G,x M T is exactly the following one: P M T ( y ) : yx π j y τ j · · · x π jt y τ jt . Thus, by definition, the following claim also holds. Claim 4 : y ∈ A ex ( H G,x , M T ) if and only if w ( y ) > w ( y τ js ) holds for all s = 1 , , · · · , t .By Claims 3 and 4, the result holds. ✷ f = ψ H ( M ) By the definition of the path P ( H,M ) ( y ), the following lemma follows. Lemma 5.1 For any y ∈ Y − Y ′ , y is adjacent to x π k on the path P ( H,M ) ( y ) if and onlyif y ∈ N H ( x π k ) − S k
For any y ∈ Y − Y ′ and ≤ k ≤ n , y ∈ D ( H, Y ′ , x π k ) if and only if y ∈ N H ( x π k ) − S k j > · · · > j t > n ( y ) ≥ n ( y τ jt ).( ⇒ ) Assume that y ∈ D ( H, Y ′ , x π k ). By Proposition 3.1 (ii), y ∈ N H ( x π k ). Let i be theminimum integer with 0 < i ≤ k such that y ∈ L ( H i ) and w ( y ) < w ( y τ i ). Such i exists26y Proposition 3.1 (ii). Thus y ∈ D ′ i . Clearly, k = n ( y ) = j and so x π k (i.e., x π j ) is thevertex on the path P ( H,M ) ( y ) adjacent to y . It remains to show that y ∈ N A ex ( H, M ).As y ∈ L ( H i ), we have q < i ≤ j , where q = n ( y ). Note that j > j > · · · > j t >n ( y ) = q ≥ n ( y τ jt ). Thus j s +1 < i ≤ j s holds for some s with 1 ≤ s ≤ t , where assumethat j t +1 = n ( y ) = q when s = t . Then y τ js ∈ L ( H i ). By Claim 1, w ( y ) < w ( y τ js ). Bydefinition, y ∈ N A ex ( H, M ).Hence the necessity holds.( ⇐ ) Now assume that y ∈ N A ex ( H, M ). Assume that j = n ( y ). We will show that y ∈ D ( H, Y ′ , x π j ).On the contrary, suppose that y / ∈ D ( H, Y ′ , x π j ). By Proposition 3.1 (ii), y / ∈ D ( H, Y ′ , x π s )for all s = 1 , , · · · , n , implying that y / ∈ D ′ i for all i = 1 , , · · · , n .As j = n ( y ) and q = n ( y ), y ∈ L ( H i ) for all i with q < i ≤ j . For each i with q < i ≤ j ,as y / ∈ D ′ i , we have w ( y ) > w ( y τ i ) by property (c). Particularly, as q < j t < · · · < j , w ( y ) > w ( y τ js ) holds for all s = 1 , , · · · , t , implying that y is externally B-active withrespect to M in H . Thus y N A ex ( H, M ), a contradiction.Hence the sufficiency holds. ✷ By Theorem 5.2 and the definition of ψ H , we have the following corollaries. Corollary 5.1 Let M ∈ U M X ( H ) . If f = ψ H ( M ) , then, f ( x π i ) is the size of the set ( N H ( x π i ) ∩ N A ex ( H, M )) − S i Let M ∈ U M X ( H ) . If f = ψ H ( M ) , then X x ∈ X f ( x ) = | N A ex ( H, M ) | ≤ | Y | − | X | . Now we apply Theorem 5.1 to find another interpretation for G-parking functions of G .Let T ∈ T ( G ). Write x π i ≪ T x π j for all i, j with 0 ≤ i < j ≤ n . For any two vertices x ′ and x in G , let P T ( x ′ , x ) denote the unique path in T between x ′ and x . Proposition 5.1 For any two different vertices x ′ and x in G , the following statementsare equivalent: (i) x ′ ≪ T x ; w max ( P T ( x ′′ , x ′ )) < w max ( P T ( x ′′ , x )) , where x ′′ is the vertex in both paths P T ( x , x ′ ) and P T ( x , x ) with E ( P T ( x ′′ , x ′ )) ∩ E ( P T ( x ′′ , x )) = ∅ ; (iii) if y is an edge in E ( G ) − E ( T ) joining x and x ′ , then x is the vertex x π j with j = n ( y ) , where π s = π s ( H G,x , E ( T )) for s ∈ { , , · · · , n } ; (iv) if y is an edge in E ( G ) − E ( T ) joining x and x ′ , then x is the vertex in the path P ( H,M ) ( y ) adjacent to y , where Y ′ = E ( T ) .Proof . Claims 1 and 2 in the proof of Theorem 5.1 imply that (i) ⇔ (ii), while thedefinition of the path P ( H,M ) ( y ) implies that (iii) ⇔ (iv). Finally, by the definition of theordering ≪ T and the definition of n ( y ), (i) ⇔ (iii) follows. ✷ Recall that the mapping φ G : T ( G ) → GP ( G, x ) is defined by φ G ( T ) = ψ H G,x ( M T ),where M T = { x π i y τ i : i = 1 , , · · · , n } by Corollary 3.2. By Corollary 5.1 and Propo-sition 5.1, we get the following interpretation for φ G which is different from the one inCorollary 4.1. Corollary 5.3 Let T ∈ T ( G ) . If f = φ G ( T ) , then, for any x ∈ V − { x } , f ( x ) is thenumber of those edges y ∈ E ( G ) − E ( T ) such that y is not externally active with respectto T in G and y is incident with x and x ′ , where x ′ ≪ T x . By Corollaries 5.2 and 5.3, we have the following conclusion. Corollary 5.4 Let T ∈ T ( G ) . If f = φ G ( T ) , then ea ( T ) + X x ∈ X f ( x ) = | E ( G ) | − | V ( G ) | + 1 , where ea ( T ) is the number of externally active edges with respect to T in G . Ω( H ; x, y, z ) Let M ∈ U M X ( H ). For any x π q ∈ X , let R ( x π q ) denote the following unique path: x π j y τ j x π j y τ j · · · x π js y τ js , where j = q , j i +1 = n ( y τ ji ) for i = 1 , , · · · , s − y τ js ∈ L ( H ). For any y ′ ∈ Y − V ( M )and r ≥ 1, if t r = n r ( y ′ ) ≥ 1, let Q r ( y ′ ) be the path in H formed by combining edge y ′ x π tr and path R ( x π tr ). In the case that y ′ ∈ L ( H ) (i.e., n ( y ′ ) < Q ( y ′ )consists of vertex y ′ only. Let k = 0 if V ( Q ( y ′ )) ∩ V ( Q ( y ′ )) ∩ X = ∅ , and let k be the28argest integer with x π k ∈ V ( Q ( y ′ )) ∩ V ( Q ( y ′ )) ∩ X otherwise. Let C H ( y ′ ) be the set { y τ u ∈ V ( Q ( y ′ )) ∪ V ( Q ( y ′ )) : u > k, y τ u = y ′ } . For example, if H is the graph in Figure 8 and M = { x i y i : i = 1 , , · · · , } , then Q ( y )is the path y x y x y x y x y x y and Q ( y ) is the path y x y x y . Thus k = 2 and C H ( y ) = { y , y , y } . For the bipartite graph H G,x and M = M T , where T ∈ T ( G ), C H G,x ( y ′ ) corresponds to the set of edges y = y ′ in the unique cycle of G [ E ( T ) ∪ { y ′ } ],where y ′ ∈ E ( G ) − E ( T ).For any y i ∈ V ( M ) ∩ Y , y i is said to be internally B-active with respect to M if w ( y i ) >w ( y ′ ) holds for each y ′ ∈ Y − V ( M ) with y i ∈ C ( y ′ ). Let A in ( H, M ) be the set of internallyB-active members with respect to M in H .Define a function Ω( H ; x, y, z ) with three variable x, y, z as follows:Ω( H ; x, y, z ) = X S ⊆ X z | X |−| S | X M ∈UM S ( H ) x ia S ( M ) y ea S ( M ) , (5.2)where ia S ( M ) = | A in ( H [ N [ S ]] , M ) | and ea S ( M ) = | A ex ( H [ N [ S ]] , M ) | .If Ω( H ; 1 , , z ) = P i ≥ c i z i , then c i is the number of members M ∈ U M ( H ) with | M | = | X | − i . In particular, c = |U M X ( H ) | .If Ω( H ; x, y, 0) = P i,j ≥ u i,j x i y j , then u i,j is the number of members M ∈ U M X ( H ) with | A in ( H, M ) | = i and | A ex ( H, M ) | = j (i.e., | N A ex ( H, M ) | = | Y | − | X | − j ).If Ω( H ; 1 , y, 0) = P j ≥ d j y j , then d j is the number of members M ∈ U M X ( H ) with | A ex ( H, M ) | = j , i.e., | N A ex ( H, M ) | = | Y | − | X | − j . By Corollary 5.2, d j is the numberof members f in BP X ( H ) with P x ∈ X f ( x ) = | Y |−| X |− j . 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