Graphs with conflict-free connection number two
Hong Chang, Trung Duy Doan, Zhong Huang, Stanislav Jendrol', Xueliang Li, Ingo Schiermeyer
aa r X i v : . [ m a t h . C O ] M a y Graphs with conflict-free connectionnumber two ∗ Hong Chang , Trung Duy Doan , † , Zhong Huang ,Stanislav Jendrol’ ‡ Xueliang Li , Ingo Schiermeyer § Center for Combinatorics and LPMCNankai University, Tianjin 300071, China Institut f¨ur Diskrete Mathematik und AlgebraTechnische Universit¨at Bergakademie Freiberg09596 Freiberg, Germany School of applied Mathematics and InformaticsHanoi University of Science and Technology, Hanoi, Vietnam Institute of Mathematics, P. J. ˇSaf´arik UniversityJesenn´a 5, 040 01 Koˇsice, SlovakiaEmail: [email protected], [email protected], [email protected]@mail.nankai.edu.cn, [email protected], [email protected]
February 12, 2018
Abstract
An edge-colored graph G is conflict-free connected if any two of its ver-tices are connected by a path, which contains a color used on exactly one of ∗ Supported by NSFC No.11531011. † Financial support by the Free State of Saxony (Landesstipendium) is thankfully acknowledged. ‡ This work was supported by the Slovak Research and Development Agency under the contractNo. APVV-15-0116 and by the Slovak VEGA Grant 1/0368/16. § Part of this research was done while the author was visiting the Center for Combinatorics.Financial support is gratefully acknowledged. ts edges. The conflict-free connection number of a connected graph G , de-noted by cf c ( G ), is the smallest number of colors needed in order to make G conflict-free connected. For a graph G, let C ( G ) be the subgraph of G inducedby its set of cut-edges. In this paper, we first show that, if G is a connectednon-complete graph G of order n ≥ C ( G ) being a linear forest and withthe minimum degree δ ( G ) ≥ max { , n − } , then cf c ( G ) = 2. The bound onthe minimum degree is best possible. Next, we prove that, if G is a connectednon-complete graph of order n ≥
33 with C ( G ) being a linear forest and with d ( x ) + d ( y ) ≥ n − for each pair of two nonadjacent vertices x, y of V ( G ), then cf c ( G ) = 2. Both bounds, on the order n and the degree sum, are tight. More-over, we prove several results concerning relations between degree conditionson G and the number of cut edges in G . Keywords: edge-coloring; conflict-free connection number; degree condition.
AMS subject classification 2010:
All graphs in this paper are undirected, finite and simple. We follow [3] for graphtheoretical notation and terminology not described here. Let G be a graph. Weuse V ( G ) , E ( G ) , n ( G ) , m ( G ), and δ ( G ) to denote the vertex-set, edge-set, number ofvertices, number of edges, and minimum degree of G , respectively. For v ∈ V ( G ), let N ( v ) denote the neighborhood of v in G , deg ( x ) denote the degree of v in G .Let G be a nontrivial connected graph with an associated edge-coloring c : E ( G ) → { , , . . . , t } , t ∈ N , where adjacent edges may have the same color. If adja-cent edges of G are assigned different colors by c , then c is a proper (edge-)coloring .For a graph G , the minimum number of colors needed in a proper coloring of G isreferred to as the edge-chromatic number of G and denoted by χ ′ ( G ). A path of anedge-colored graph G is said to be a rainbow path if no two edges on the path have thesame color. The graph G is called rainbow connected if every pair of distinct verticesof G is connected by a rainbow path in G . An edge-coloring of a connected graph isa rainbow connection coloring if it makes the graph rainbow connected. This conceptof rainbow connection of graphs was introduced by Chartrand et al.[7] in 2008. Fora connected graph G , the rainbow connection number rc ( G ) of G is defined as thesmallest number of colors that are needed in order to make G rainbow connected.Readers interested in this topic are referred to [17, 18, 19] for a survey.Inspired by the rainbow connection coloring and the proper coloring in graphs,2ndrews et al.[1] and Borozan et al.[4] independently introduced the concept of aproper connection coloring. Let G be a nontrivial connected graph with an edge-coloring. A path in G is called a proper path if no two adjacent edges of the pathreceive the same color. An edge-coloring c of a connected graph G is a proper con-nection coloring if every pair of distinct vertices of G is connected by a proper pathin G . And if k colors are used, then c is called a proper connection k -coloring . Anedge-colored graph G is proper connected if any two vertices of G are connected bya proper path. For a connected graph G , the minimum number of colors that areneeded in order to make G proper connected is called the proper connection number of G , denoted by pc ( G ). Let G be a nontrivial connected graph of order n and size m (number of edges). Then we have that 1 ≤ pc ( G ) ≤ min { χ ′ ( G ) , rc ( G ) } ≤ m . Formore details, we refer to [2, 13, 14, 15] and a dynamic survey [16].Our research was motivated by the following three results. Theorem 1.1 [5] If G is a 2-connected graph of order n = n ( G ) and minimum degree δ ( G ) > max { , n +820 } , then pc ( G ) ≤ . Theorem 1.2 [5] For every integer d ≥ , there exists a 2-connected graph of order n = 42 d such that pc ( G ) ≥ . Theorem 1.3 [14] Let G be a connected noncomplete graph of order n ≥ . If G / ∈ { G , G } and δ ( G ) ≥ n , then pc ( G ) = 2 , where G and G are two exceptionalgraphs on 7 and 8 vertices.A coloring of the vertices of a hypergraph H is called conflicted-free if each hy-peredge E of H has a vertex of unique color that is not repeated in E . The smallestnumber of colors required for such a coloring is called the conflict-free chromaticnumber of H . This parameter was first introduced by Even et al. [12] in a geometricsetting, in connection with frequency assignment problems for cellular networks. Onecan find many results on the conflict-free coloring, see [9, 10, 20].Recently, Czap et al. [8] introduced the concept of a conflict-free connection ofgraphs. An edge-colored graph G is called conflict-free connected if each pair ofdistinct vertices is connected by a path which contains at least one color used onexactly one of its edges. This path is called a conflict-free path , and this coloring iscalled a conflict-free connection coloring of G . The conflict-free connection number of a connected graph G , denoted by cf c ( G ), is the smallest number of colors neededto color the edges of G so that G is conflict-free connected. In [8], they showed thatit is easy to compute the conflict-free connection number for 2-connected graphs andvery difficult for other connected graphs, including trees.3his paper is organized as follows. In Section 2, we list some fundamental resultson the conflict-free connection of graphs. In Sections 3 and 4, we prove our mainresults. At the very beginning, we state some fundamental results on the conflict-freeconnection of graphs, which will be used in the sequel.
Lemma 2.1 [8] If P n is a path on n edges, then cf c ( P ) = ⌈ log ( n + 1) ⌉ . Let C ( G ) be the subgraph of G induced on the set of cut-edges of G . The follow-ing lemmas respectively provide a necessary condition and a sufficient condition forgraphs G with cf c ( G ) = 2.Recall that a linear forest is a forest where each of its components is a path. Lemma 2.2 [8] If cf c ( G ) = 2 for a connected graph G , then C ( G ) is a linear forestwhose each component has at most three edges. Lemma 2.3 [8] If G is a connected graph, and C ( G ) is a linear forest in which eachcomponent is of order , then cf c ( G ) = 2 . The following lemma, which can be seen as a corollary of Lemma 2.3 for C ( G )being empty, is of extra interest. A rigorous proof can be found in [11]. Lemma 2.4 [8, 11] If G is a -edge-connected non-complete graph, then cf c ( G ) = 2 . A block of a graph G is a maximal connected subgraph of G that has no cut-vertex. If G is connected and has no cut-vertex, then G is a block. An edge is a blockif and only if it is a cut-edge, this block is called trivial . Therefore, any nontrivialblock is 2-connected. Lemma 2.5 [8]
Let G be a connected graph. Then from its every nontrivial block anedge can be chosen so that the set of all such chosen edges forms a matching. Let C ( G ) be a linear forest consisting of k ( k ≥
0) components Q , Q , . . . , Q k with n i = | V ( Q i ) | such that 2 ≤ n ≤ n ≤ · · · ≤ n k . We now present a strongerresult than Lemma 2.3, which will be important to show our main results.4 heorem 2.6 If G is a connected non-complete graph with C ( G ) being a linear forestwith n = n = · · · = n k − ≤ n k ≤ or C ( G ) being edgeless, then cf c ( G ) = 2 .Proof. If C ( G ) is edgeless then the theorem is true by Lemma 2.4. If C ( G ) alinear forest with at least one edge, then G is a non-complete graph and therefore cf c ( G ) ≥
2. It remains to verify the converse. Note that one can choose from eachnontrivial block an edge so that all the chosen edges create a matching set S byLemma 2.5. We define an edge-coloring of G as follows. First, we color all edgesfrom S with color 2, and the edges in E ( G ) \ { S ∪ Q k } with color 1. Next, we onlyneed to color the edges of Q k . If n k = 2, then color the unique edge of Q k with color1. If n k = 3, then color two edges of Q k with colors 1 and 2. Suppose n k = 4. Itfollows that Q k is a path of order 4, say w w w w . We color the two edges w w and w w with color 1, and w w with color 2. It is easy to check that this coloringis a conflict-free connection coloring of G . Thus, we have cf c ( G ) ≤
2, and hence cf c ( G ) = 2. Remark 1:
The following example points out that Theorem 2.6 is optimal in senseof the number of components with more than two vertices of the linear forest C ( G )of a graph G .For t ≥
3, let S n be the graph with n = 5 t vertices, consisting of the path P = v v v v v v with complete graphs K t attached to the vertices v i , i ∈ { , , , } and one more K t sharing the edge v v with P . Observe that δ ( S n ) = t − n − ,and C ( S n ) is a linear forest with two components of order 3, paths v v v and v v v .In any conflict-free connection coloring of S n with two colors the edges v v and v v (resp. v v and v v ) receive different colors. But then any v - v path has a conflict.This means that cf c ( S n ) ≥ Theorem 3.1
Let G be a connected graph of order n ≥ k , k ≥ . If δ ( G ) ≥ n − k +1 k ,then G has at most k − cut edges.Proof. Assume or the sake of contradiction that G has at least k − B be a set of k − G . Then the graph G \ B has exactly k components G , . . . , G k . Consider the following two cases.Case 1. For every j ∈ [ k ] there is a vertex v j ∈ V ( G j ) such that N ( v j ) ⊆ V ( G j ).5hen every component G j has at least n − k +1 k + 1 vertices and we have n = | V ( G ) | = k X j =1 | V ( G j ) | ≥ k · ( n − k + 1 k + 1) = n + 1 , a contradiction.Case 2. There exists some i ∈ [ k ] such that N ( v ) V ( G i ) for every vertex v ∈ V ( G i ) . Then a = | V ( G i ) | ≤ k − v ∈ V ( G i ) is incident witha cut edge from B. Let m i denote the degree sum of all the vertices of V ( G i ) within G [ V ( G i ) ∪ B ]. Then we have n − k + 1 k · a ≤ m i ≤ a · ( a −
1) + k − . This, together with the bounds on a , provides0 ≤ a · ( a − − n − k + 1 k ) + k − ≤ ( k − · ( k − − n − k + 1 k ) + k − . This leads to n ≤ k − , a contradiction.The next theorem shows that the bound on the minimum degree in Theorem 3.1cannot be lowered. Theorem 3.2
For every k ≥ and t ≥ there exists a connected n -vertex graph H n with n = k · t , δ ( H n ) = n − kk , and k − cut edges.Proof. The graph H n consists of a path P k on k vertices to every vertex of it acomplete graph K t is attached.The following theorem shows that the bound k on the number n of vertices inTheorem 3.1 is best possible. Theorem 3.3
For every k ≥ there exists a graph R n on n = k − vertices with δ ( R n ) = n − k +1 k and k − cut edges.Proof. The graph R n is a connected graph consisting of a central block B , isomor-phic to the complete graph K k − , k − B , . . . , B k − , that are complete graphson k vertices, and a matching M of k − B with the remaining blocks. Theorem 3.4
Let G be a connected graph of order n ≥ max { k + k, ⌊ k ⌋ · k ( k −
2) + k − k + 3 k − } , k ≥ . f deg( x ) + deg( y ) ≥ n − k +1 k for any two non-adjacent vertices x and y of G , then G has at most k − cut edges.Proof. Assume for the sake of contradiction that G has at least k − B be a set of k − G . Then the graph G \ B has exactly k components G , . . . , G k . Consider the following two cases.Case 1. For every j ∈ [ k ] there is a vertex v j ∈ V ( G j ) such that N ( v j ) ⊆ V ( G j ).Case 1.1. Let k be even. Then n = | V ( G ) | = k X j =1 | V ( G j ) ∪ V ( G k − j +1 ) | ≥ k · ( 2 n − k + 1 k + 2) = n + 12 , a contradiction.Case 1.2. Let k be odd. Then, w.l.o.g., we can suppose that | V ( G k ) | ≥ n − k +1 k + 1.Therefore, n = | V ( G k ) | + k − X j =1 | V ( G j ) ∪ V ( G k − j ) | ≥ n − k + 1 k + 1 + k − · ( 2 n − k + 1 k + 2)= n + k + 12 k , a contradiction.Case 2. There exists some i ∈ [ k ] such that N ( v ) V ( G i ) for every vertex v ∈ V ( G i ) . Case 2.1. There exists only one i ∈ [ k ] such that all vertices v ∈ V ( G i ) have N ( v ) V ( G i ). Observe that | V ( G i ) | = a ≤ k −
1. Notice that every vertex v ∈ V ( G i ) is incident with an edge from B , and there is a vertex y ∈ V ( G i ) withdeg( y ) ≤ a − k − a . For any component G j , j = i ∈ [ k ], there is | V ( G j ) | ≥ ⌈ n − k + 1 k ⌉ − deg( y ) + 1 ≥ ⌈ n − k + 1 k ⌉ − a + 1 − k − a + 1 . This means that the number of vertices in G is n = | V ( G ) | ≥ ( k − · ( ⌈ n − k + 1 k ⌉ − a + 1 − k − a + 1) + a ≥ ( k − · ( 2 n − k + 1 k − a + 1 − k − a + 1) + a. After some manipulations we get n ≤ k ( k − k − a · k − k − k − a − k ) . a , provides n ≤ k ( k − k − · k − k − k − − k ) . The inequality yields n ≤ k + k + 1 k − . Next we check whether n = k + k satisfies the original inequality n = | V ( G ) | ≥ ( k − · ( ⌈ n − k + 1 k ⌉ − a + 1 − k − a + 1) + a. After some manipulations we get k + k ≥ k + 2 k − , which is impossible. Then we have n ≤ k + k − , a contradiction.Case 2.2 There exists more than one i ∈ [ k ] such that all vertices v ∈ V ( G i ) have N ( v ) V ( G i ). Assume that there exists a pair of non-adjacent vertices u, w with u ∈ V ( G i ) and w ∈ V ( G i ) . It is possible that i = i . Notice that every vertex insuch a component is incident with an edge from B , and the two vertices u and w areincident with at most one edge from B in common, then deg ( u ) + deg ( w ) − ≤ k − n ≤ k +2 k − , a contradiction. Now we get that every vertex in such com-ponents is adjacent to the remaining vertices of such components. Hence all possibleconfigurations have been excluded except for two adjacent singletons { u } , { w } as theonly such two components V i , V i . As deg( u ) + deg( w ) − ≤ k −
1, w.l.o.g., weassume that deg ( u ) ≤ ⌊ k ⌋ . For any component G j , j = i or i , then | V ( G j ) | ≥ n − k + 1 k − deg( u ) + 1 ≥ n − k + 1 k − ⌊ k ⌋ + 1 . This means that the number of vertices in G is n = | V ( G ) | ≥ ( k − · ( 2 n − k + 1 k − ⌊ k ⌋ + 1) + 2 . After some manipulations we get n ≤ ⌊ k ⌋ · k ( k −
2) + k − k + 2 k − , Remark 2:
Observe that the graph H n of Theorem 3.2 is a good example showingthat the bound on the sum of degrees in Theorem 3.4 is tight.The next theorem shows that the bound on n cannot be lower than k + k . Theorem 3.5
For every k ≥ there exists a graph D n on n = k + k − verticeswith deg( x ) + deg( y ) ≥ n − k +1 k for any two non-adjacent vertices x and y and having k − cut edges.Proof. Let D n be a graph consisting of a vertex v , k − B , . . . , B k − , thatare complete graphs on k + 2 vertices, and a set M of k − v with the k − B , . . . , B k − . Observe that D n is a connectedgraph on k + k − x ) + deg( y ) ≥ k ≥ n − k +1 k for any twonon-adjacent vertices x and y . cf c ( G ) = 2 Theorem 4.1
Let G be a connected non-complete graph of order n ≥ . If C ( G ) induces a linear forest and δ ( G ) ≥ n − , then cf c ( G ) = 2 .Proof. Observe that, by Theorem 3.1, the subgraph C ( G ) of any connected graph G with δ ( G ) ≥ n − contains at most three cut edges. As C ( G ) is a linear forest, weconclude that cf c ( G ) = 2 by Theorem 2.6. Remark 3:
The graph S n defined in the end of Section 2 provides a good exampleshowing the tightness of the minimum degree in Theorem 4.1.Next, we discuss the minimum degree condition for small graphs to have conflict-free connection number 2. Theorem 4.2
Let G be a connected non-complete graph of order n , ≤ n ≤ . If C ( G ) induces a linear forest and δ ( G ) ≥ max { , n − } , then cf c ( G ) = 2 .Proof. We may assume that C ( G ) = ∅ by Lemma 2.4. Let C ( G ) consist of k components Q , Q , . . . , Q k with n i = | V ( Q i ) | such that 2 ≤ n ≤ n ≤ · · · ≤ n k . Wemay also assume that 3 ≤ n k − ≤ n k ≤ G \ ( E ( Q k − ) ∪ E ( Q k )) has at least five components C , C , C , C , C . Since δ ( G ) ≥ | V ( C i ) | > ≤ i ≤
5. Notice that at most two vertices in C i can be contained in Q k − ∪ Q k , then for each C i there exists a vertex u i such that N ( u i ) ⊆ V ( C i ) for 1 ≤ i ≤
5. Thus, | V ( G ) | ≥ P i =1 | V ( C i ) | ≥ P i =1 ( d ( u i ) + 1) ≥ n − + 1) = n + 1 > n , a contradiction, which completes the proof. Remark 4:
The following examples show that the minimum degree condition inTheorem 4.2 is best possible. Let H i be a complete graph of order three for 1 ≤ i ≤ v i of H i for 1 ≤ i ≤
2. Let H be a graph obtained from H , H by connecting v and v with a path of order t for t ≥
5. Note that δ ( H ) = 2, but cf c ( H ) ≥
3. Another graph class is given as follows. Let G i be a complete graph oforder n , and take a vertex w i of G i for 1 ≤ i ≤
5. Let G be a graph obtained from G , G , G , G , G by joining w i and w i +1 with an edge for 1 ≤ i ≤
4. Notice that δ ( G ) = n − , but cf c ( G ) ≥ Theorem 4.3
Let G be a connected noncomplete graph of order n with ≤ n ≤ .If C ( G ) induces a linear forest and δ ( G ) ≥ , then cf c ( G ) = 2 .Proof. If | E ( C ( G )) | ≤
3, then the proof follows from Theorem 2.6. Otherwise thesubgraph G \ E ( C ( G )) has at least five components. Since δ ( G ) ≥
2, at least twocomponents of it have at least three vertices. Thus | V ( G ) | ≥ × >
8, acontradiction.
Remark 5.
The following example shows that the minimum degree condition inTheorem 4.3 is best possible. Let G be a path of order t with t ≥
5. It is easy to seethat δ ( G ) = 1, but cf c ( G ) = ⌈ log t ⌉ ≥ C ( G ) is a linear forest in above theorems, then we canget the following theorem. Theorem 4.4
Let G be a connected non-complete graph of order n ≥ . If δ ( G ) ≥ n − , then cf c ( G ) = 2 .Proof. Observe that Theorem 3.1 shows that C ( G ) of any connected graph G with δ ( G ) ≥ n − has at most two edges. This, when applying Theorem 2.6, immediatelygives our theorem. Remark 6:
The following example shows that the minimum degree condition inTheorem 4.4 is best possible. Let H i be a complete graph of order n for 1 ≤ i ≤
4, andtake a vertex v i of H i for 1 ≤ i ≤
4. Let H be a graph obtained from H , H , H , H by adding the edges v v , v v , v v . Note that δ ( H ) = n − , but cf c ( H ) ≥
3. On10he other hand, the condition n ≥
16 in Theorem 4.4 is also best possible. Let G , G , G , G be complete graphs of order 1 , , ,
5, respectively, and take a vertex w i of G i for 1 ≤ i ≤
4. Let G be a graph obtained from G , G , G , G by adding theedges w w , w w , w w . Note that δ ( G ) ≥ n − , but cf c ( G ) ≥
3. Also the graph R from Theorem 3.3 shows the sharpness of the bound of n . Theorem 4.5
Let G be a connected non-complete graph of order n ≥ . If C ( G ) isa linear forest, and deg( x ) + deg( y ) ≥ n − for each pair of two non-adjacent vertices x and y of V ( G ) , then cf c ( G ) = 2 .Proof. From Theorem 3.4 we deduce that the subgraph C ( G ) of G has at most threeedges. Now the proof follows from Theorem 2.6. Remark 7:
An example of the graph S n , introduced in Remark 1, shows thatthe degree sum condition in Theorem 4.5 is best possible. On the other hand, thecondition n ≥
33 in Theorem 4.5 is also best possible. Let G i be a complete graph oforder n − for 1 ≤ i ≤ n ≤
32, and G = v u u v v be a path of order 5. Let G be a graph obtained from G , G , G , G by identifying a vertex of G i to the vertex v i for 1 ≤ i ≤
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