aa r X i v : . [ m a t h . C O ] O c t On hamiltonian colorings of trees
Devsi Bantva
Lukhdhirji Engineering College, Morvi - 363 642Gujarat (INDIA) [email protected]
Abstract. A hamiltonian coloring c of a graph G of order n is a map-ping c : V ( G ) → { , , , ... } such that D ( u, v ) + | c ( u ) − c ( v ) | ≥ n −
1, forevery two distinct vertices u and v of G , where D ( u, v ) denotes the de-tour distance between u and v which is the length of a longest u, v -pathin G . The value hc(c) of a hamiltonian coloring c is the maximum colorassigned to a vertex of G . The hamiltonian chromatic number, denotedby hc ( G ), is the min { hc ( c ) } taken over all hamiltonian coloring c of G . Inthis paper, we present a lower bound for the hamiltonian chromatic num-ber of trees and give a sufficient condition to achieve this lower bound.Using this condition we determine the hamiltonian chromatic number ofsymmetric trees, firecracker trees and a special class of caterpillars. Keywords:
Hamiltonian coloring, hamiltonian chromatic number, sym-metric tree, firecracker, caterpillar. A hamiltonian coloring c of a graph G of order n is a mapping c : V ( G ) →{ , , , ... } such that D ( u, v ) + | c ( u ) − c ( v ) | ≥ n −
1, for every two distinct vertices u and v of G , where D ( u, v ) denotes the detour distance between u and v whichis the length of a longest u , v -path in G . The value of hc ( c ) of a hamiltoniancoloring c is the maximum color assigned to a vertex of G . The hamiltonianchromatic number hc ( G ) of G is min { hc ( c ) } taken over all hamiltonian coloring c of G . It is clear from definition that two vertices u and v can be assignedthe same color only if G contains a hamiltonian u , v -path, and hence a graph G can be colored by a single color if and only if G is hamiltonian-connected.Thus the hamiltonian chromatic number of a connected graph G measures howclose G is to being hamiltonian-connected. The concept of hamiltonian coloringwas introduced by Chartrand et al. [2] which is a variation of radio k -coloring ofgraphs.At present, the hamiltonian chromatic number is known only for handfulof graph families. Chartrand et al. [2,3] determined the hamiltonian chromaticnumber for complete graph K n , cycle C n , star K ,k , complete bipartite graph K r,s and presented upper bound for the hamiltonian chromatic number of pathsand trees. The exact value of hamiltonian chromatic number of paths which isequal to the radio antipodal number ac ( P n ) was given by Khennoufa and Togni Devsi Bantva in [6]. Shen et al. [7] discussed the hamiltonian chromatic number for graphs G with max { D ( u, v ) : u, v ∈ V ( G ), u = v } ≤ n/
2, where n is the order of graph G ;such graphs are called graphs with maximum distance bound n/ DB ( n/ A tree is a connected graph that contains no cycle. The diameter of T ,denoted by diam ( T ) or simply d , is the maximum distance among all pairs ofvertices in T . The eccentricity of a vertex in a graph is the maximum distancefrom it to other vertices in the graph, and the center of a graph is the set ofvertices with minimum eccentricity. It is well known that the center of a tree T ,denoted by C ( T ), consists of a single vertex or two adjacent vertices, called the central vertex/vertices of T . We view T as rooted at its central vertex/vertices;if T has only one central vertex w then T is rooted at w and if T has twoadjacent central vertices w and w ′ then T is rooted at w and w ′ in the sensethat both w and w ′ are at level 0. If u is on the path joining another vertex v and central vertex w , then u is called ancestor of v , and v is a descendent of u .Let u C ( T ) be adjacent to a central vertex. The subtree induced by u and allits descendent is called a branch at u . Two branches are called different if theyare at two vertices adjacent to the same central vertex, and opposite if they areat two vertices adjacent to different central vertices. Define the detour level of avertex u from the center of graph by L ( u ) := min { D ( u, w ) : w ∈ C ( T ) } , u ∈ V ( T ) . Define the total detour level of T as L ( T ) := X u ∈ V ( T ) L ( u ) . For any u, v ∈ V ( T ), define φ ( u, v ) := max {L ( t ) : t is a common ancestor of u and v } , and δ ( u, v ) := , if C ( T ) = { w, w ′ } and path P uv contains an edge ww ′ , , otherwise. n hamiltonian colorings of trees 3 Lemma 1.
Let T be a tree with diameter d ≥ . Then for any u, v ∈ V ( T ) thefollowing holds:1. φ ( u, v ) ≥ ;2. φ ( u, v ) = 0 if and only if u and v are in different or opposite branches;3. δ ( u, v ) = 1 if and only if T has two central vertices and u and v are inopposite branches;4. the detour distance D ( u, v ) in T between u and v can be expressed as D ( u, v ) = L ( u ) + L ( v ) − φ ( u, v ) + δ ( u, v ) . (1)Note that for a tree T the detour distance D ( u, v ) is same as the ordinarydistance d ( u, v ) as there is unique path between any two vertices u and v of T .Thus, one can use expression (1) for ordinary distance d ( u, v ) which can also beused for other purpose.Define ε ( T ) := , if C ( T ) = { w } , , if C ( T ) = { w, w ′ } .ε ′ ( T ) := 1 − ε ( T ) . For a connected graph G of order n ≥
5, by defining D ( σ ) = P n − i =1 D ( v i , v i +1 )for an ordering σ : v , v , .... , v n and D ( G ) = max { D ( σ ) : σ is an ordering of V ( G ) } , Chartrand et al. [4] established the following lower bound for the hamil-tonian chromatic number of a connected graph G . Theorem 1. ([4]) If G is a connected graph of order n ≥ , then hc ( G ) ≥ ( n − + 1 − D ( G ) . For an ordering σ : v , v , .... , v n of the vertices of G , define c σ to be anassignment of positive integers to V ( G ): c σ ( v ) = 1 and c σ ( v i +1 ) − c σ ( v i ) =( n − − D ( v i , v i +1 ) for each 1 ≤ i ≤ n −
1. If max { D ( u, v ) : u, v ∈ V ( G ) , u = v }≤ n/ G of order n then such a graph G is called agraph with maximum distance bound n/ DB ( n/
2) graph for short. Shen et al. [7] proved the following Theorems about DB ( n/
2) graphs and using itdetermined the hamiltonian chromatic number for double stars and a specialclass of caterpillars.
Theorem 2. ([7]) Let G be a DB ( n/ graph of order n ≥ . Then for any σ , c σ is a hamiltonian coloring for G with hc ( c σ ) = ( n − + 1 − D ( σ ) . Devsi Bantva
Theorem 3. ([7]) If G is DB ( n/ graph of order n ≥ , then hc ( G ) = ( n − + 1 − D ( G ) , and for any σ such that D ( σ ) = D ( G ) , hc( c σ ) = hc( G ). Namely, c σ is a minimum hamiltonian coloring for G . Now, let T be a tree with maximum degree ∆ . Note that a hamiltoniancoloring c of T is injective for ∆ ( T ) ≥ T containhamiltonian path. Throughout this section we consider T with ∆ ( T ) ≥ c induces a linear order of the vertices of T , namely V ( T ) = { u , u , ..., u n − } (where n = | V ( T ) | ) such that0 = c ( u ) < c ( u ) < ... < c ( u n − ) = span( c ). Theorem 4.
Let T be a tree of order n ≥ and ∆ ( T ) ≥ . Then hc ( T ) ≥ ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ) . (2) .Proof. It is enough to prove that any hamiltonian coloring of T has span notless than the right-hand side of (2). Suppose c is any hamiltonian coloring of T then c order the vertices of T into a linear order u , u ,..., u n − such that 0= c ( u ) < c ( u ) < ... < c ( u n − ). By definition of c , we have c ( u i +1 ) − c ( u i ) ≥ n − − D ( u i , u i +1 ) for 0 ≤ i ≤ n −
1. Summing up these n − c ) = c ( u n − ) ≥ ( n − − n − X i =0 D ( u i , u i +1 ) (3) Case-1: T has one central vertex. In this case, we have φ ( u i , u i +1 ) ≥ δ ( u i , u i +1 ) = 0 for 0 ≤ i ≤ n − φ and δ . Since T has only one central vertex, u and u n − cannot be the central vertex of T simultaneously. Hence L ( u ) + L ( u n − ) ≥
1. Thus, by substituting (1) in (3),span( c ) ≥ ( n − − n − X i =0 [ L ( u i ) + L ( u i +1 ) − φ ( u i , u i +1 ) + δ ( u i , u i +1 )]= ( n − − n − X i =0 L ( u i ) + L ( u ) + L ( u n − ) − n − X i =0 φ ( u i , u i +1 ) ≥ ( n − + 1 − L ( T )= ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ) . Case-2: T has two central vertices. In this case, we have φ ( u i , u i +1 ) ≥ δ ( u i , u i +1 ) ≤ ≤ i ≤ n − φ and δ . Since T has two central vertices, we can set { u , u n − } = { w, w ′ } . Thus, by substituting(1) in (3),span( c ) ≥ ( n − − n − X i =0 [ L ( u i ) + L ( u i +1 ) − φ ( u i , u i +1 ) + δ ( u i , u i +1 )] n hamiltonian colorings of trees 5 = ( n − − n − X i =0 [ L ( u i ) + L ( u i +1 )] − n − X i =0 φ ( u i , u i +1 ) + n − X i =0 δ ( u i , u i +1 )= ( n − − n − X i =0 L ( u i ) + L ( u ) + L ( u n − ) + n − X i =0 δ ( u i , u i +1 ) ≥ ( n − − X u ∈ V ( T ) L ( u i ) + ( n − n − n − − L ( T )= ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ) . Theorem 5.
Let T be a tree of order n ≥ and ∆ ( T ) ≥ . Then hc ( T ) = ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ) (4) holds if there exists a linear order u , u ,..., u n − with 0 = c ( u ) < c ( u ) < ... 2. Then c ( u j ) − c ( u i ) = j − X t = i [ c ( u t +1 ) − c ( u t )]= j − X t = i [ n − − L ( u t ) − L ( u t +1 ) − ε ( T )]= j − X t = i [ n − − D ( u t , u t +1 )] Devsi Bantva = ( j − i )( n − − j − X t = i D ( u t , u t +1 ) ≥ ( j − i )( n − − ( j − i ) (cid:16) n (cid:17) = ( j − i ) (cid:18) n − (cid:19) ≥ n − D ( u i , u j ) ≥ 1; it follows that | c ( u j ) − c ( u i ) | + D ( u i , u j ) ≥ n − 1. Hence, c is a hamiltonian coloring for T . The span of c is given byspan( c ) = c ( u n − ) − c ( u )= n − X t =0 [ c ( u t +1 ) − c ( u t )]= n − X t =0 [ n − − L ( u t ) − L ( u t +1 ) − ε ( T )]= ( n − − n − X t =0 [ L ( u t ) + L ( u t +1 )] − ( n − ε ( T )= ( n − n − − ε ( T )) − L ( T ) + L ( u ) + L ( u n − )= ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T )Therefore, hc ( T ) ≤ ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ). This together with(2) implies (4) and that c is an optimal hamiltonian coloring. Corollary 1. Let T be a DB ( n/ tree (or d ≤ n/ ) of order n ≥ and ∆ ( T ) ≥ , where d is diameter of T . Then hc ( T ) = ( n − n − − ε ( T )) + ε ′ ( T ) − L ( T ) (7) holds if there exists a linear order u , u ,..., u n − with 0 = c ( u ) < c ( u ) < ... 1. A tree is said to be a caterpillar C if it consists of a path v v ...v m ( m ≥ C , with some hanging edges known as legs, which are incidentto the inner vertices v , v ,..., v m − . If d ( v i ) = k for i = 2 , , ..., m − 1, then wedenote the caterpillar by C ( m, k ), where d ( v i ) denotes the degree of v i . For allabove defined trees it is easy to verify that d ≤ n/ 2, and hence DB ( n/ 2) treesas max { D ( u, v ): u, v ∈ V ( T ) } ≤ d ≤ n/ u , u , ..., u n − of vertices of T which satisfies conditions of Corollary 1. Theorem 6. Let k, d ≥ be integers. Then hc ( T k +1 ( d ))= ( k +1) ( k − ( k d − h ( k d − 1) + k +1 (2 − ( k − d ) i − k +1 k − d + 1 , if d is even , k ( k − ( k d − − h k ( k d − − 1) + 1 i + kk − (2 − d ) k d − − kk − , if d is odd. (10) Proof. Note that T k +1 ( d ) has one or two central vertex/vertices depending on d and hence we consider the following two cases. Case 1 : d is even.In this case T k +1 ( d ) has a unique central vertex, denoted by w . Denote thechildren of the central vertex w by w , w , . . . , w k +1 . Denote the k children ofeach w t by w t , w t , . . . , w tk − , 1 ≤ t ≤ k + 1. Denote the k children of each w ti by w ti , w ti , . . . , w ti ( k − , 0 ≤ i ≤ k − 1, 1 ≤ t ≤ k + 1. Inductively, denote the k children of w ti ,i ,...,i l (0 ≤ i , i , . . . , i l ≤ k − 1, 1 ≤ t ≤ k + 1) by w ti ,i ,...,i l ,i l +1 where 0 ≤ i l +1 ≤ k − 1. Continue this until all vertices of T k +1 ( d ) are indexedthis way. We then rename the vertices of T k +1 ( d ) as follows:For 1 ≤ t ≤ k + 1, set v tj := w ti ,i ,...,i l , where j = 1 + i + i k + · · · + i l k l − + X l +1 ≤ t ≤⌊ d/ ⌋ k t . Devsi Bantva We give a linear order u , u , . . . , u n − of the vertices of T k +1 ( d ) as follows.We first set u = w . Next, for 1 ≤ j ≤ n − k − 2, let u j := ( v ts , where s = ⌈ j/ ( k + 1) ⌉ , if j ≡ t (mod ( k + 1)) for t with 1 ≤ t ≤ k,v k +1 s , where s = ⌈ j/ ( k + 1) ⌉ , if j ≡ k + 1)) . Finally, let u j := w j − n + k +2 , n − k − ≤ j ≤ n − . Note that u n − = w k +1 is adjacent to w , and for 1 ≤ i ≤ n − u i and u i +1 arein different branches so that φ ( u i , u i +1 ) = 0. Case 2 : d is odd.In this case T k +1 ( d ) has two (adjacent) central vertices, denoted by w and w ′ . Denote the neighbours of w other than w ′ by w , w , . . . , w k − and the neigh-bours of w ′ other than w by w ′ , w ′ , . . . , w ′ k − . For 0 ≤ i ≤ k − 1, denote the k children of each w i (respectively, w ′ i ) by w i , w i , . . . , w i ( k − (respectively, w ′ i , w ′ i , . . . , w ′ i ( k − ). Inductively, for 0 ≤ i , i , . . . , i l ≤ k − 1, denote the k children of w i ,i ,...,i l (respectively, w ′ i ,i ,...,i l ) by w i ,i ,...,i l ,i l +1 (respectively, w ′ i ,i ,...,i l ,i l +1 ), where 0 ≤ i l +1 ≤ k − 1. We rename v j := w i ,i ,...,i l , v ′ j := w ′ i ,i ,...,i l , where j = 1+ i + i k + · · · + i l k l − + X l +1 ≤ t ≤⌊ d/ ⌋ k t . We give a linear order u , u , . . . , u n − of the vertices of T k +1 ( d ) as follows. Wefirst set u := w, u n − := w ′ , and for 1 ≤ j ≤ n − 2, let u j := ( v s , where s = ⌈ j/ ⌉ , if j ≡ v ′ s , where s = ⌈ j/ ⌉ , if j ≡ . Then u i and u i +1 are in opposite branches for 1 ≤ i ≤ n − 2, and u i +2 j , j =0,1,...,( k − 1) are in different branches for 1 ≤ i ≤ n − k + 1, so that φ ( u i , u i +1 ) = 0 and δ ( u i , u i +1 ) = 1.Therefore, in each case above, a defined linear order of vertices satisfies theconditions of Corollary 1. The hamiltonian coloring defined by (8) and (9) is anoptimal hamiltonian coloring whose span equal to the right-hand side of (7). Butit is straight forward to verify that the order of T k +1 ( d ) is given by n := k +1 k − ( k d − , if d is even , (cid:16) kk − ( k d − − (cid:17) , if d is odd . (11) n hamiltonian colorings of trees 9 With the help of formula 1 + 2 x + 3 x + ... + px p − = px p x − − x p − x − , one canverify that the total level of T k +1 ( d ) is given by L ( T k +1 ( d )) := ( k + 1) (cid:18) dk d k − − k d − k − (cid:19) , if d is even2 k (cid:18) ( d − k d − k − − k d − − k − (cid:19) , if d is odd . (12)By substituting (11) and (12) into (7), we obtain the right-hand side of (10) isthe hamiltonian chromatic number of T k +1 ( d ). Theorem 7. For m ≥ and k ≥ , hc ( F ( m, k )) = m k − m ( k − − k ( m − 1) + 2 , if m is odd ,m k − m ( k − − k m + 2 , if m is even. (13) Proof. Let w i , w i ,..., w ik denote the vertices of the i th copy of the ( k − F ( m, k ), where w i is the apex vertex (center) and w i ,..., w ik are the leaves.Without loss of generality we assume that w k , w k ,..., w mk are identified to thevertices in the path of length m − F ( m, k ). Note that F ( m, k ) has one or two central vertex/vertices depending on m and hence weconsider the following two cases. Case-1: m is odd.In this case F ( m, k ) has only one central vertex w which is w ⌊ m ⌋ k . We give alinear order u , u ,..., u n − of the vertices of F ( m, k ) as follows. We first set u = w = w ⌊ m ⌋ k . Next, for 1 ≤ t ≤ n − m , let u t := w ij , where t = ( j − m + ( i − ⌊ m ⌋ ) , if i = ⌊ m ⌋ ( j − m + 2 i, if i < ⌊ m ⌋ ( j − m + 2( i − ⌊ m ⌋ ) + 1 , if i > ⌊ m ⌋ . Finally, for n − m + 1 ≤ t ≤ n − 1, let u t := w ij , where t = ( ( j − m − i − ⌊ m ⌋ ) + 1 , if i < ⌊ m ⌋ ( j − m + 2( m − i + 1) , if i > ⌊ m ⌋ . Case-2: m is even.In this case F ( m, k ) has two central vertices w and w ′ which are w m k and w m +1 k respectively. We give a linear order u , u ,..., u n − of the vertices of F ( m, k ) as follows. We first set u = w ′ = w m +1 k and u n − = w = w m k . Next,for 1 ≤ t ≤ n − m + 1, let u t := w ij , where t = ( ( j − m + 2 i − , if i ≤ m ( j − m + 2( i − m ) , if i > m . Finally, for n − m + 2 ≤ t ≤ n − 2, let u t := w ij , where t = ( ( j − m + 2 i − , if i < m ( j − m + 2( i − − m ) , if i > m + 1 . Therefore, in each case above, a defined linear order of vertices satisfies con-ditions of Corollary 1. The hamiltonian coloring defined by (8) and (9) is anoptimal hamiltonian coloring whose span equal to the right-hand side of (7).But the order and total level of firecrackers F ( m, k ) are given by n := mk (14) L ( F ( m, k )) := km +(8 k − m − k , if m is odd , km +6 m ( k − , if m is even. (15)By substituting (14) and (15) into (7), we obtain the right-hand side of (13) isthe hamiltonian chromatic number of F ( m, k ). Theorem 8. Let m, k ≥ . Then hc ( C ( m, k ))= ( ( m − k − (5 m − m + 19) k + (3 m − m + 11) , if m is odd , ( m − k − (5 m − m + 20) k + (3 m − m + 12) , if m is even. (16) Proof. Let v , v ,..., v m be the vertices of spine and v ji , 1 ≤ j ≤ k − i th , 2 ≤ i ≤ m − C ( m, k ) has one or twocentral vertex/vertices depending on m and hence we consider the following twocases. Case-1: m is odd.In this case C ( m, k ) has only one central vertex which is v ⌊ m ⌋ = w . We firstset u = v ⌊ m ⌋ +1 , u n − = w and other vertices as follows.For 1 ≤ t ≤ m − u t := v i , where t = ( i − , if i < ⌊ m ⌋ , i − ⌊ m ⌋ ) , if i > ⌊ m ⌋ + 1.For m − ≤ t ≤ n − u t := v ji , where t = ( m − j + 2( i − , if i < ⌊ m ⌋ , ( m − j + 1 , if i = ⌊ m ⌋ , ( m − j + 2( i − ⌊ m ⌋ ) + 1 , if i > ⌊ m ⌋ . Case-2: m is even.In this case C ( m, k ) has two central vertices which are v m = w and v m +1 = w ′ . We first set u = v m +1 , u n − = m and other vertices as follows. n hamiltonian colorings of trees 11 For 1 ≤ t ≤ m − u t := v i , where t = ( i − , if i < m − , i − m ) , if i > m + 1.For m − ≤ t ≤ n − u t := v ji , where t = ( ( m − j + 2( i − 2) + 1 , if i ≤ m , ( m − j + 2( i − m ) , if i > m .Therefore, in each case above, a defined linear order of vertices satisfies con-ditions of Corollary 1. The hamiltonian coloring defined by (8) and (9) is anoptimal hamiltonian coloring whose span equal to the right-hand side of (7).But the order and total level of caterpillars C ( m, k ) are given by n := m ( k − − k − 2) (17) L ( C ( m, k )) := ( m − k − + 1 , if m is odd , m ( m − k − , if m is even. (18)By substituting (17) and (18) into (7), we obtain the right-hand side of (16) isthe hamiltonian chromatic number of C ( m, k ).We remark that Theorem 5 is also useful to determine hamiltonian chromaticnumber of non DB ( n/ 2) trees. See the following result. Theorem 9. Let P ′ m be a tree obtained by attaching a pendant vertex to centralvertex/vertices of path P m . Then hc ( P ′ m ) := ( m − , if m is odd , m + 2 m − , if m is even . (19) Proof. The order and total level of P ′ m are given by n := ( m + 1 , if m is odd ,m + 2 , if m is even . (20) L ( P ′ m ) := m +34 , if m is odd , m − m +84 , if m is even . (21)Substituting (20) and (21) into (2) we obtain that the right-hand side of (19) isa lower bound for hc ( P ′ m ). Now we give a linear ordering of vertices of P ′ m whichsatisfies conditions of Theorem 5. Note that P ′ m has one central vertex when m is odd and two adjacent central vertices when m is even. Hence we consider thefollowing two cases. Case-1: m is odd.Let v v ...v m be the vertices of path and v ′ be the vertex attached to centralvertex v ( m +1) / then we order the vertices as follows: v ( m +1) / , v , v ( m +3) / , v , v ( m +5) / , v , v ( m +7) / , ...., v ( m − / , v m , v ′ . Rename the vertices of P ′ m in the above ordering by u , u ,..., u n − . Namely, let u = v ( m +1) / , u = v ,..., u n − = v ′ then it satisfies conditions of Theorem 5. Case-2: m is even.Let v v ...v m be the vertices of path and v ′ and v ′′ are attached to centralvertices v m/ and v m/ then we order the vertices as follows: v m/ , v , v m/ , v , v m/ , v , ...., v m/ − , v m , v ′ , v ′′ , v m/ . Rename the vertices of P ′ m in the above ordering by u , u ,..., u n − . Namely, let u = v m/ , u = v ,..., u n − = v m/ then it satisfies conditions of Theorem 5.Therefore, in each case above, a defined linear order of vertices of P ′ m satisfiesconditions of Theorem 5 and hence the hamiltonian coloring defined by (5) and(6) is an optimal hamiltonian coloring whose span is (4) which is (19) for thecurrent case. Acknowledgement I want to express my deep gratitude to an anonymous referee for kind commentsand constructive suggestions.