Berge-Fulkerson coloring for infinite families of snarks
aa r X i v : . [ m a t h . C O ] A ug Berge-Fulkerson coloring for infinite families of snarks
Ting Zheng, Rong-Xia Hao ∗ Department of Mathematics, Beijing Jiaotong University,Beijing, 100044, P.R.ChinaSeptember 25, 2018
Abstract
It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has sixperfect matchings such that each edge is contained in exactly two of them. H¨ a gglundconstructed two graphs Blowup( K , C ) and Blowup( P rism, C ). Based on these twographs, Chen constructed infinite families of bridgeless cubic graphs M , , ,...,k − ,k − which is obtained from cyclically 4-edge-connected and having a Fulkerson-cover cubicgraphs G , G , . . . , G k − by recursive process. If each G i for 1 ≤ i ≤ k − M , , , has a Fulkerson-cover and gave the open problem that whether every graph in M , , ,...,k − ,k − has aFulkerson-cover. In this paper, we solve this problem and prove that every graph in M , , ,...,k − ,k − has a Fulkerson-cover. Keyword : Berge-Fulkerson conjecture, Fulkerson-cover, perfect matching.
MSC(2010) : Primary: 05C70; Secondary: 05C75, 05C40, 05C15.
Let G be a simple graph with vertex set V ( G ) and edge set E ( G ). A circuit of G is a2-regular connected subgraph. An even graph is a graph with even degree at every vertex.A perfect matching of G is a 1-regular spanning subgraph of G . The excessive index of G ,denoted by χ ′ e ( G ), is the least integer k , such that G can be covered by k perfect matchings.A cubic graph is a snark if it is bridgeless and not 3-edge-colorable. A cubic graph G is Berge-Fulkerson colorable if 2 G is 6-edge-colorable. It is an equivalent description of theBerge-Fulkerson conjecture.The following is a famous open problem called Berge-Fulkerson conjecture. ∗ Corresponding author, [email protected] onjecture 1.1. (Berge-Fulkerson Conjecture [4], or see [10]) Every bridgeless cubic graphhas six perfect matchings such that each edge belongs to exactly two of them.
Although there are some results related with this conjecture, as examples, see [2],[5],[7],[9],[8], Berge-Fulkerson conjecture is still open for many bridgeless cubic graphs even forsome simple snarks.H¨ a gglund [7] constructed two graphs Blowup( K , C ) and Blowup( P rism, C ). Based onBlowup( K , C ) , Esperet et al. [3] constructed infinite families of cyclically 4-edge-connectedsnarks with excessive index at least five. Based on these two graphs, Chen [1] constructedinfinite families of cyclically 4-edge-connected snarks E , , ,..., ( k − obtained from cyclically4-edge-connected snarks G , G , . . . , G k − , in which E , , is Esperet et al.’ construction. Ifonly assume that each graph in { G , G , . . . , G k − } has a Fulkerson-cover, then these infinitefamilies of bridgeless cubic graphs are denoted by M , , ,...,k − ,k − . Chen [1] obtained thatevery graph in M , has a Fulkerson-cover and each graph in M , , , has a Fulkerson-coverand gave the following problem. Problem 1.2. [1] If H = { G ; G , G , . . . , G k − , G k − } ∈ M , , ,...,k − ,k − , Does H have aFulkerson-cover? In this paper, we solve Problem 1.2. The main result is Theorem 1.3.
Theorem 1.3.
Each graph in M , , ,...,k − ,k − for k > has a Fulkerson-cover. In this section, some necessary definitions, constructions and Lemma are given.Let X ⊆ V ( G ) and Y ⊆ E ( G ). We use G \ X to denote the subgraph of G obtainedfrom G by deleting all the vertices of X and all the edges incident with X . While G \ Y to denote the subgraph of G obtained from G by deleting all the edges of Y . The edge-cutof G associated with X , denoted by ∂ G ( X ), is the set of edges of G with exactly one endin X . The edge set C = ∂ G ( X ) is called a k -edge-cut if | ∂ G ( X ) | = k . A cycle of G is asubgraph of G with each vertex of even degree. A circuit of G is a minimal 2-regular cycleof G . A graph G is called cyclically k -edge-connected if at least k edges must be removedto disconnect it into two components, each of which contains a circuit.Let G i be a cyclically 4-edge-connected snark with excessive index at least 5, for i = 0 , x i y i be an edge of G i and x i , x i ( y i , y i ) be the neighbours of x i ( y i ). Let H i be thegraph obtained from G i by deleting the vertices x i and y i . Let { G ; G , G } be the graphobtained from the disjoint union of H , H by adding six vertices a , b , c , a , b , c and23 edges a y , a x , a c , c b , b y , b x , b x , b y , b c , c a , a x , a y , c c . Thegraphs of this type are denoted as E , (see Figure 1). H H a c b a b c Figure 1: { G ; G , G } The family of graphs E , ,..., ( k − ( k ≥
2) and M , ,..., ( k − ( k ≥
2) are constructed byChen as follows:(1) { G ; G , G } ∈ E , with A j = { a j , b j , c j } for j = 0 , ≤ i ≤ k , { G ; G , G , . . . , G i − } is obtained from { G ; G , G , . . . , G i − } ∈ E , ,..., ( i − by adding H i − and A i − = { a i − , b i − , c i − } and by inserting a vertex v i − into e , such that(i) G i − is a cyclically 4-edge-connected snark with excessive index at least 5 ( x i − y i − is an edge of G i − and x i − , x i − ( y i − , y i − ) are the neighbours of x i − ( y i − ));(ii) H i − = G i − \{ x i − , y i − } ;(iii) e ∈ E ( { G ; G , G , . . . , G i − } ) − i − S j =0 E ( H j ) − i − S j =0 { a j c j , c j b j } and e is incident with c ; (iv) a i − is adjacent to x and y i − , b i − is adjacent to x and y i − , a i − is adjacent to x i − and y i − , b i − is adjacent to x i − and y i − , c i − is adjacent to a i − , b i − and v i − , theother edges of { G ; G , G , . . . , G i − } remain the same;(v) { G ; G , G , . . . , G i − } ∈ E , ,..., ( i − .The class of graphs constructed by Esperet et al. is a special case for k = 3 of E , ,..., ( k − .If the excessive index and non 3-edge-colorability of G i ( i = 0 , , , . . . , ( k − G i has a Fulkerson-cover, then we obtain infinite families of bridgelesscubic graphs. We denote graphs of this type as M , , ,..., ( k − ( k ≥ Lemma 2.1. (Hao, Niu, Wang, Zhang and Zhang [6])
A bridgeless cubic graph G has aFulkerson-cover if and only if there are two disjoint matchings E and E , such that E ∪ E is a cycle and G \ E i is 3-edge colorable, for each i = 0 , . Each graph in M , , ,...,k − ,k − has a Fulkerson cover We give the results according to the parity of k . Theorem 3.1.
Let k be an even integer and k ≥ . If Γ ∈ M , , ,...,k − ,k − , then Γ has aFulkerson-cover.Proof. Since Γ ∈ M , , ,...,k − ,k − , assume Γ = { G ; G , G , . . . , G k − , G k − } .Since G i has a Fulkerson-cover, for each i = 0 , , . . . , k −
1, suppose that { M i , M i , M i , M i , M i , M i } is the Fulkerson-cover of G i . Let E i be the set of edges in G i covered twiceby { M i , M i , M i } and E i be the set of edges in G i which are not covered by { M i , M i , M i } .Note that E i ∪ E i is an even cycle, and G i \ E i and G i \ E i can be colored by three colors.Then E i and E i are the desired disjoint matchings of G i as in Lemma 2.1. By choosingthree perfect matchings of G i , for each i = 0 , , . . . , k −
1, we can obtain two desired disjointmatchings E i and E i such that x i y i ∈ E i ∪ E i or x i , y i V ( E i ∪ E i ).Three perfect matchings { M i , M i , M i } of G i are chosen such that x i , y i V ( E i ∪ E i )if i is even; And three perfect matchings { M i , M i , M i } of G i are chosen such that x i y i ∈ E i ∪ E i if i is odd. Without loss of generality, assume that x i y i ∈ E i and x i x i , y i y i ∈ E i for odd i .Let E = ( E − x y ) ∪ ( E − { x x , y y } ) ∪ k − [ i =2 ( E i − x i y i ) ∪ k − [ i =2 a i c i ∪ k − [ j =1 v j − v j ∪ { c v k − , c v , y a , x a } and E = ( E − { x x , y y } ) ∪ ( E − x y ) ∪ k − [ i =2 ( E i − { x i x i , y i y i } ) ∪ k − [ j =1 { y j +1 a j +1 , x j +1 a j } ∪ k − [ i =2 v i − c i ∪ { a c , a c } . Clearly, E ∪ E is an even cycle C . (See Figure 2.)If i is odd, by x i y i ∈ E i , there exists a maximal path containing only 2-degree vertices asinter vertices in the graph G i \ E i , say u i . . . y i x i . . . u i , which corresponds to an edge u i u i in the graph G i \ E i (see Figure 3). From x i x i , y i y i ∈ E i , there exists two maximal pathscontaining only 2-degree vertices as inter vertices in the graph G i \ E i , say u i . . . x i x i x i . . . u i and u i . . . y i y i y i . . . u i , which corresponds to an edge u i u i and u i u i , respectively, in thegraph G i \ E i (see Figure 4). 4igure 2: { G ; G , G , . . . , G k − } If i is even, by x i , y i V ( E i ), there exist four maximal paths containing only 2-degreevertices as inter vertices in the graph G i \ E i , say u i . . . x i x i (maybe u i = x i ), u i . . . x i x i (maybe u i = x i ), u i . . . y i y i (maybe u i = y i ) and u i . . . y i y i , which correspond to fouredges u i x i , u i x i , u i y i and u i y i , respectively, in the graph G i \ E i (see Figure 5).Similarly, by x i , y i V ( E i ), there exist four maximal paths containing only 2-degreevertices as inter vertices in the graph G i \ E i , say u i . . . x i x i (maybe u i = x i ), u i . . . x i x i (maybe u i = x i ), u i . . . y i y i (maybe u i = y i ) and u i . . . y i y i (maybe u i = y i ), whichcorrespond to four edges u i x i , u i x i , u i y i and u i y i , respectively, in G i \ E i (see Figure 6).From the construction of Γ, we know that Γ \ E (see Figure 7) is5 i x i y i x i y i u i u i G i \ E i u i u i G i \ E i Figure 3 G i x i y i x i y i u i u i G i \ E i u i u i G i \ E i u i u i u i u i Figure 4( G \ E − u u ) ∪ k − S j =0 ( G j \ E j − { x j , y j } ) ∪ k − S j =1 ( G j +1 \ E j +1 − { u j +1 u j +1 , u j +1 u j +1 } ) ∪ { u b , u b , b u , b u , u b , u b }∪ k − S j =1 { u j u j +1 , u j b j , u j +1 b j , u j +1 u j +2 , u j +1 b j +1 , u j +2 b j +1 , b j b j +1 } and Γ \ E (see Figure 7) is( G \ E − { u u , u u } ) ∪ k − S j =0 ( G j \ E j − { x j , y j } ) ∪ k − S j =1 ( G j +1 \ E j +1 − u j +1 u j +1 ) ∪{ u b , u u , b u , u b , u u , u b , b b }∪ k − S j =1 { u j b j , u j b j , b j u j +1 , u j +1 b j +1 , b j +1 u j +2 , b j +1 u j +2 } . If i is odd, because E i and E i are the desired disjoint matchings of G i as in Lemma 2.1, G i \ E i is 3-edge colorable. Thus there exists a 2-factor, say C i , such that each component6 i x i y i x i y i u i u i G i \ E i u i u i G i \ E i u i u i u i u i y i x i Figure 5 G i x i y i x i y i u i u i G i \ E i u i u i G i \ E i u i u i u i u i y i x i Figure 6is an even circuit and u i u i is not in the 2-factor C i . Similarly, because G i \ E i is 3-edge colorable, there exists a 2-factor C i such that each component is an even circuit and { u i u i , u i u i } is in the 2-factor C i .If i is even, because G i \ E i is 3-edge colorable, there exists a 2-factor C i such thateach component is an even circuit and u i x i u i and u i y i u i are in the 2-factor C i . Because G i \ E i is 3-edge colorable, there exists a 2-factor C i such that each component is an evencircuit and { u i x i u i , u i y i u i } is in the 2-factor C i .Then Γ \ E has a 2-factor: C ∪ k − S j =1 ( C j +1 − { u j +1 u j +1 , u j +1 u j +1 } ) ∪ k − S j =0 ( C j − { x j , y j } ) ∪ { b u , b u ,b u , b u } ∪ k − S j =1 { b j u j +1 , b j u j , u j u j +1 , b j +1 u j +1 , b j +1 u j +2 , u j +1 u j +2 } . And each component is an even circuit. 7igure 7: { G ; G , G , . . . , G k − } \ E and { G ; G , G , . . . , G k − } \ E And Γ \ E has a 2-factor: k − S j =1 C j +1 ∪ ( C − { u u , u u } ) ∪ k − S j =0 ( C j − { x j , y j } ) ∪ { b u , b u , u u ,b u , u u , b u } ∪ k − S j =1 { u j b j , u j b j , u j +2 b j +1 , u j +2 b j +1 } . And each component is an even circuit.So Γ \ E and Γ \ E are 3-edge colorable. Therefore E and E are the desired match-ings in Γ of Lemma 2.1. So Γ = { G ; G , G , . . . , G k − , G k − } has a Fulkerson-cover. Theorem 3.2.
Let k be an odd integer. If H ∈ M , , ,...,k − ,k − , then H has a Fulkerson-cover.Proof. By H ∈ M , , ,...,k − ,k − , assume H = { G ; G , G , . . . , G k − , G k − } . Since G i hasa Fulkerson-cover, for each i = 0 , , . . . , k −
1, suppose that { M i , M i , M i , M i , M i , M i } is the Fulkerson-cover of G i . Let E i be the set of edges covered twice by { M i , M i , M i } and E i be the set of edges not covered by { M i , M i , M i } . Now E i ∪ E i is an even cycle,and G i \ E i and G i \ E i can be colored by three colors. Then E i and E i are the desireddisjoint matchings of G i as in Lemma 2.1. By choosing three perfect matchings of G i , foreach i = 0 , , . . . , k −
1, we can obtain two desired disjoint matchings E i and E i such that x i y i ∈ E i ∪ E i or x i , y i V ( E i ∪ E i ). If i is even and i = 0, three perfect matchings of G i x i , y i V ( E i ∪ E i ). If i is odd or i = 0, three perfect matchings of G i are chosen such that x i y i ∈ E i ∪ E i . Without loss of generality, assume that x i y i ∈ E i and x i x i , y i y i ∈ E i for i = 0 or i is odd.Let E = ( E − x y ) ∪ ( E − { x x , y y } ) ∪ k − [ i =2 ( E i − x i y i ) ∪ { y a , x a , c v }∪ k − [ i =2 a i c i ∪ k − [ j =0 v j +1 v j +2 and E = ( E − x y ) ∪ k − [ i =2 ( E i − { x i x i , y i y i } ) ∪ ( E − { x x , y y } ) ∪{ a c } ∪ k − [ i =1 { a i x i +1 , a i +1 y i +1 } ∪ k − [ i =2 { v i − c i } . (See Figure 8). Clearly, E ∪ E is an even cycle C .Figure 8: { G ; G , G , . . . , G k − , G k − } If i is odd or i = 0, by x i y i ∈ E i , there exists a maximal path containing only 2-degreevertices as inter vertices in the graph G i \ E i , say u i . . . y i x i . . . u i , which corresponds to anedge u i u i in the graph G i \ E i (see Figure 3); By x i x i , y i y i ∈ E i , there exists two distinctmaximal path containing only 2-degree vertices as inter vertices in the graph G i \ E i , say9 i . . . x i x i x i . . . u i and u i . . . y i y i y i . . . u i which correspond to the edges u i u i and u i u i respectively in the graph G i \ E i (see Figure 4).If i is even and i = 0, since x i , y i V ( E i ), there exist four maximal paths containingonly 2-degree vertices as inter vertices in the graph G i \ E i , say u i . . . x i x i (maybe u i = x i ), u i . . . x i x i (maybe u i = x i ), u i . . . y i y i (maybe u i = y i ) and u i . . . y i y i (maybe u i = y i ),which correspond to the edges u i x i , u i x i , u i y i and u i y i , respectively, in the graph G i \ E i .(See Figure 5). Similarly, by x i , y i V ( E i ), there exist four maximal paths containing only2-degree vertices as inter vertices in the graph G i \ E i , say u i . . . x i x i (maybe u i = x i ), u i . . . x i x i (maybe u i = x i ), u i . . . y i y i (maybe u i = y i ) and u i . . . y i y i which correspondto the edges u i x i , u i x i , u i y i and u i y i ,respectively, in the graph G i \ E i .If k = 1, then H = G which has a Fulkerson-cover.If k ≥
2, we will prove H \ E and H \ E are 3-edge colorable in the following.From the construction of H , one has that H \ E (see Figure 9) is( G \ E − { u u , u u } ) ∪ ( G \ E − u u ) ∪ Q ∪ Q ∪ S k − j =1 { u j b j , u j +1 b j , u j +1 u j }∪ k − S j =1 { u j +1 b j +1 , u j +1 u j +2 , u j +2 b j +1 , b j b j +1 } ∪ { u c , u b , b c , b k − c , u b , u b , u b , u b } . Where, Q = k − S j =1 ( G j \ E j −{ x j , y j } ) and Q = k − S j =1 ( G j +1 \ E j +1 −{ u j +1 u j +1 , u j +1 u j +1 } )And H \ E (see Figure 9) is( G \ E − { u u , u u } ) ∪ ( G \ E − u u ) ∪ k − S j =1 ( G j \ E j − { x j , y j } ) ∪ k − S j =1 ( G j +1 \ E j +1 − { u j +1 u j +1 } ) ∪ S k − j =1 { u j b j , u j +1 b j , u j b j } ∪ k − S j =1 { u j +1 b j +1 , u j +2 b j +1 , u j +2 b j +1 }∪{ u u , u b , u b , u c , u b , b c , u b , c b } . If i is odd or i = 0, because G i \ E i is 3-edge colorable, there exists a 2-factor C i suchthat each component is an even circuit and u i u i is not in the 2-factor C i . Because G i \ E i is 3-edge colorable, there exists a 2-factor C i such each component is an even circuit and u i u i , u i u i are in the 2-factor C i .If i is even and i = 0, because G i \ E i is 3-edge colorable, there exists a 2-factor C i such each component is an even circuit and two paths with length two u i x i u i and u i y i u i are in the 2-factor C i . Similarly, because G i \ E i is 3-edge colorable, there exists a 2-factor C i such each component is an even circuit and u i x i u i , u i y i u i are in the 2-factor C i .10hen H \ E (see Figure 9) has a 2-factor:( C − { u u , u u } ) ∪ C ∪ k − S j =1 ( C j +1 − { u j +1 u j +1 , u j +1 u j +1 } ) ∪ k − S j =1 ( C j − { x j , y j } ) ∪ k − S j =1 { b j u j +1 , b j u j , u j u j +1 } ∪ k − S j =1 { b j +1 u j +1 ,b j +1 u j +2 , u j +1 u j +2 } ∪ { b u , b u , b u , b c , u c } and each component is an even circuit.Figure 9: H \ E and H \ E H = { G ; G , G , . . . , G k − , G k − } H \ E has a 2-factor: C ∪ ( C − { u u , u u } ) ∪ k − S j =1 ( C j − { x j , y j } ) ∪ k − S j =1 C j +1 ∪ S k − j =1 { u j b j , u j b j }∪ k − S j =1 { u j +2 b j +1 , u j +2 b j +1 } ∪ { c u , b c , b u , u u , u b , u b } . And each component is an even circuit.So H \ E and H \ E are 3-edge colorable. Therefore E and E are the desired match-ings of Lemma 2.1 and H = { G ; G , G , . . . , G k − , G k − } has a Fulkerson-cover.11rom Theorem 3.1 and Theorem 3.2, we get the Theorem 1.3 that every graph in M , , ,..., ( k − has a Fulkerson-cover. Acknowledgments
This work was supported by the NSFC (No.11371052), the Fundamental Research Fundsfor the Central Universities (Nos. 2016JBM071, 2016JBZ012), the 111 Project of China(B16002).
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