- Email: [email protected]
Note on the perfect EIC-graphs
Applied Mathematics and Computation 289 (2016) 481–485
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Note on the perfect EIC-graphs Jun Yue a, Shiliang Zhang b,∗, Xia Zhang a a b
School of Mathematical Sciences, Shandong Normal University, Jinan 250014, Shandong, China Mechanical and Electrical Engineering Department, Jinan Engineering Vocational Technical College, Jinan 250200, Shandong, China
a r t i c l e
i n f o
Keywords: Edge coloring Injective coloring Injective edge coloring Injective edge chromatic index Perfect EIC-graph
a b s t r a c t Three edges e1 , e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length 3. The injective edge coloring number χi (G ) is the minimum number of colors permitted in a coloring of the edges of G such that if e1 , e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. Let ω denote the number of edges in a maximum clique of G. A graph G is called an ω edge injective colorable (or perfect EIC-)graph if χi (G ) = ω . In this paper, we give a sharp bound of the injective coloring number of a 2-connected graph with some forbidden conditions, and then we also characterize some perfect EIC-graph classes, which extends the results of perfect EIC-graph of Cardoso et al. in [Injective edge chromatic index of a graph, http://arxiv.org/abs/1510.02626.]. © 2016 Elsevier Inc. All rights reserved.
1. Introduction All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty [1]. Except the classical vertex coloring and edge coloring, there are many kinds of colorings are studied, such as list coloring, star coloring and acyclic coloring. In addition, rainbow connection and rainbow vertex-connection are the new ones, we refer to a survey [10] and some new papers [8,9]. Recently, a new edge coloring–injective edge coloring, which has concrete applications in network science, causes researchers’ attentions. In this paper, we will focus on it. An injective coloring of G is a coloring of the vertices of G such that for every vertex v ∈ V (G ), all the neighbors of v are assigned distinct colors, i.e., if x and y are two distinct neighbors of v, then c(x) = c(y). The smallest integer k such that G has an injective k-coloring is the injective chromatic number of G, denoted by χ i (G). Injective coloring of graphs was introduced by Hahn et al. in [7] and was originated from Complexity Theory on Random Access Machines, and can be applied in the theory of error correcting codes [7]. In the same paper, they proved that, for k ≥ 3, it is NP-complete to decide whether the injective chromatic number of a graph is at most k. Since then, many researchers studied on this coloring and got many beautiful results. For more details, we refer to [2,3,5,6,11,12]. Similar to the injective coloring, an edge version of the injective coloring was introduced by Cardoso et al. in [4]. An injective edge coloring of a graph G is an edge coloring of G such that if e1 , e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. The injective edge coloring number or injective edge chromatic index of a graph G, denoted by χi (G ), is the minimum number of colors permitted in an injective edge coloring of G. In [4], Cardoso et al. gave the exact values of injective edge coloring number for several classes of graphs, such as path, complete bipartite graph, complete
∗
Corresponding author. Tel.: +86 15966063925. E-mail addresses: [email protected] (J. Yue), [email protected] (S. Zhang), [email protected] (X. Zhang).
http://dx.doi.org/10.1016/j.amc.2016.05.031 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
482
J. Yue et al. / Applied Mathematics and Computation 289 (2016) 481–485
Fig. 1. A block graph G and a friendship graph Fn .
graph and so on. And further, they also gave some bounds on injective edge coloring number of some graph and proved that checking whether χi (G ) = k is NP -complete. The clique number of a graph G, denoted by ω(G), is the number of vertices in a maximum clique of G. The number of edges in a maximum clique of G is denoted by ω (G). Obviously, ω (G ) = ω (G )(ω2(G )−1 ) . It is easy to see that χ i (G) ≥ ω(G) and χi (G ) ≥ ω (G ). We say that G is an ω edge injective colorable (perfect EIC-)graph if χi (G ) = ω . In [4], the authors gave some perfect EIC-graphs, such as the complete graph, the star, the friendship graph and the unicyclic graph with K3 . However, there are many unknown perfect EIC-graphs. In the same paper, the authors also gave the following open problem, which is stated as follows. Problem 1.1. How to characterize the perfect EIC-graph, i.e., how to use the structure of the graph to characterize the graph with χi (G ) = ω (G ). In this paper, we consider the injective edge coloring of graphs. We then determine injective edge colorings and injective edge coloring numbers. In the next section, we give some basic definitions and concepts, which will be used in the following sections. And then in Section 3, we give a sharp bound of a 2-connected graph by using ear decomposition, and also we construct a 2-connected perfect EIC-graph class. Section 4 gives the construction of a graph which is an perfect EIC-graph with cut-vertex. 2. Preliminaries In this section, we firstly start with some basic concepts and definitions. And then we list some known results, which will be used in the following sections. For other notation and terminology, we refer to [1] and [13]. Let G = (V, E ) be a graph. The number of vertices of a graph G is its order, written as n; its number of edges is its size, denoted by m. The degree d (v ) of a vertex v in G is the number of edges at v. If d (v ) = 1, the vertex v is called a pendant vertex. And the edge is called a pendant edge if one of its vertices is a pendant vertex. As usually, the path, the cycle, and the complete graph with n vertices will be denoted by Pn , Cn and Kn , respectively. A vertex v is called a cut-vertex in G if G − v contains more components than G does; in particular if G is connected, then a cut-vertex is a vertex v such that G − v is disconnected. Similarly, a bridge (or cutedge) is an edge whose deletion increases the number of components. A block in a graph G is a maximal nonseparable subgraph—that is, a nonseparable subgraph that is not properly contained in any other nonseparable subgraph of G. A nonseparable graph is itself often called a block. K2 is a block, but obviously no other block can contain a bridge. The clique is a subgraph of G such that it is a complete graph. The clique number ω(G) is the number of vertices in a maximum clique of G. The number of edges in a maximum clique of G is denoted by ω (G). If G has size m ≥ 1, then ω (G ) = ω (G )(2ω−1 ) . Three edges e1 , e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length 3. A block graph (or clique tree) G is a graph in which every biconnected component (block) is a clique (see Fig. 1). A block of a graph G containing only one cut vertex is called an end block of G. A friendship graph Fn is a planar graph with 2n + 1 vertices and 3n edges, i.e., the graph with 2n + 1 vertices formed by n ≥ 1 triangles all attached to a common vertex (see Fig. 1). Now, we list some known results about the perfect EIC-graphs, which are stated as follows. Lemma 2.1 ([4]). For any connected graph G of order n ≥ 2, χi (G ) ≥ ω (G ). Example 2.1 ([4]). The following graphs are examples of perfect EIC-graphs. 1. The complete graph Kn ; 2. The start K1,q ; 3. The friendship graph Fn . Before giving the following proposition, we need a operation cornoa, which is stated as follows. The cornoa Kp ◦ K1 is the graph obtained from Kp by adding a pendant edge to each of its vertices.
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Proposition 2.1 ([4]). For any positive integer p ≥ 3, consider the complete graph Kp with V (K p ) = {v1 , . . . , v p }, and a family of stars K1,q1 , . . . , K1,q p , with qj ≥ 1. Let G be the graph obtained coalescing a maximum degree vertex of the star K1,q j with the vertex v j of the Kp , for j = 1, . . . , p. Then G is a perfect EIC-graph. Similarly to the graph of the above proposition, we present a construction of the perfect EIC-graph as follows. Lemma 2.2. For any positive integer n ≥ 1, consider the friendship graph Fn whose common vertex is v and triangles are labeled with T1 , T2 , . . . , Tn and a family of complete graphs Kq1 , Kq2 , . . . , Kqs with 0 ≤ s ≤ n. Let G be a graph obtained by replacing the triangle Ti in Fn by the complete graph Ki , where 1 ≤ i ≤ s. Then G is a perfect EIC-graph. Proof. Let qi and mi be the order and size of the complete graph Kqi , where 1 ≤ i ≤ s. Suppose that m1 ≤ m2 ≤ ≤ ms . By Lemma 2.1, we can get χi (G ) ≥ ms , since the complete graph Kqs is a subgraph of G . Now, we only to show that there exists a coloring c of G using ms colors to make G be an injective edge coloring graph. We color the edges of G as follows. For Kqs , color each edge incidenting with v with a different color in [1, ns ], and then color the other edges with different colors in [ns + 1, ms ]; for Kqi , color each edges incidenting with v by a different color in [1, ni ], and then color the other edges in Kqi with different colors in [ms + ni − mi , ms ]. It is easy to check that this coloring c is an injective edge coloring of G . The result is obtained. Note that if s = n and all the complete graphs are K2 , then G is a star K1,n ; if s = n and all the complete graphs are K3 , then G is the friendship graph. To end this section, we listed the following known result about the unicyclic graphs. Proposition 2.2 ([4]). If G is a unicyclic graph with K3 , then G is a perfect EIC-graph. 3. Injective edge coloring of 2-connected graph We start this section with a famous graph decomposition, named ear decomposition, of a 2-connected graph, which can be used to characterize several important classes of graphs. Let H be a subgraph of a graph G. An ear of H in G is a nontrivial path in G whose end vertices lie in H but whose internal vertices are not. An ear decomposition of a 2-connected graph G is a sequence G0 , G1 , . . . , Gk of 2-connected subgraph of G such that (1) G0 is a cycle of G; (2) Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of Gi−1 in G; (3) Gi−1 (1 ≤ i ≤ k) is a proper subgraph of Gi ; (4) Gk = G. Now, we give the following lemma which will be used in this section. Lemma 3.1. Let H be a graph with injective edge chromatic number χi (H ) ≥ (H ) + 1. If Pn = v0 , v1 , . . . , vn is an ear of H and v0 , vn ∈ V (H ), n ≥ 4 or n = 2, then χi (G ) ≤ χ (H ) + 1, where G = H ∪ Pn . Proof. Let c: E(H) → [k] be an injective edge coloring of H. Let c[v] be the set of colors of the edges incidenting with v ∈ V (H ). Since χi (Pn ) = 2, for n ≥ 4 and χi (H ) ≥ + 1, we can color the Pn using a new color k + 1 and a color k0 of c \ c[v0 ] and a color kn of c \ c[vn ]. Now, we color the graph G as follows. First, keep the injective edge coloring of H. And then color the edges v0 v1 , vn−1 vn with the color k + 1, and color the leave edges of Pn with k0 , kn . We can check that it is an injective edge coloring of G. Then χi (G ) ≤ χ (H ) + 1 holds. Similarly to the above lemma, we give the following observation. Observation 3.1. Let H be a graph with injective edge chromatic number χi (H ) ≥ (H ) + 1. If Pn = v0 , v1 , . . . , v3 is an ear of H and v0 , v3 ∈ V (H ), then χi (G ) ≤ χ (H ) + 2, where G = H ∪ Pn . Using the above results, we get a bound of injective edge coloring of a 2-connected graph G, which can be stated as follows. Theorem 3.1. Let G be a 2-connected graph. And G0 , G1 , . . . , Gk be an ear decomposition of G such that Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of length at least 2 of Gi−1 in G, then
χi (G ) =
2k + 2
if |G0 | = 0 (mod 4 ),
2k + 3
otherwise.
Furthermore, if Pi−1 is an ear of length at least 2 of Gi−1 in G and χi (Gi ) ≥ (Gi ) + 1, the number of ears of length 3 is t, then χi (G ) = k + t + 3. And the bounds are sharp. Proof. The proof of the theorem is easily got from Lemma 3.1 and Observation 3.1, since χi (Pn ) = 2, for n ≥ 4. Now we give the examples to show the bounds are sharp. The graph G is obtained from a cycle C2k+1 with an ear of length of t ≥ 4. It is easily to check that the injective edge coloring of G must use χi (C2k+1 ) + 1 colors. Then the theorem holds. Now, we construct a graph G∗ as follows. Let G0 = Kn where n ≥ 5; and Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of Gi−1 in G, and the degree of the vertices except the vertices in Kn is at most n2 for every Gi , where 1 ≤ i ≤ k; then G∗ = Gk .
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Fig. 2. The graph Kn2 .
Theorem 3.2. The graph G∗ is a perfect EIC-graph. Proof. Because the degree of the vertices except V(Kn ) is at most G∗ ,
then χi (G∗ ) ≥ n(n2−1 ) .
n 2,
Kn is the maximum clique of G∗ . Since Kn is an included
graph of Now, we give an injective edge coloring c of G∗ as follows. First, we color each edge of G0 with a different color in n (n−1 ) [ 2 ]. For 1 ≤ i ≤ k, color the ear Pi of Gi by the following cases. If two end vertices of Pi are in G0 , i.e., Pi = v0 , v1 , . . . , vs and v0 , vs ∈ V (G0 ), then we color the ear Pi using two colors which are colored the edges v01 vs1 , v02 vs2 , where v0i ∈ NG0 (v0 ) and vsi ∈ NG0 (vs ), i ∈ {1, 2}. If two end vertices of Pi = v0 , v1 , . . . , vs are not in G0 , then there must exist a color k0 that not used by the edges which can be form the consecutive edges with v0 v1 in Gi , since d (v0 ) ≤ 2n . Similarly, there also exists a color ks that is not used by the edges which can be form the consecutive edges with vs−1 vs in Gi . Note that d (v0 ) and d (vs ) ≤ n2 , then k0 = ks . If on end vertex of Pi is in G0 and another is not in G0 , i.e., v0 ∈ V (G0 ). Since d (vs ) ≤ n2 , then there must be two different colors to color the ear Pi such that Gi an injective edge coloring graph. Then we can check that the coloring c of G∗ an injective edge coloring. Then the theorem holds. 4. Perfect EIC-graph with cut-vertex In this section, we first give some basic lemmas, which will be used in the following section. And then we construct a class of ω EIC-graphs, which extends the results in [4]. Now, we begin this section with a special graph which is not a perfect EIC-graph. Let Kn2 be a graph obtained by connecting Kn and Kn with an edge e0 = vv , where v ∈ V (Kn ), v ∈ V (Kn ), (see Fig. 2). Observation 4.1. Kn2 is a n(n2−1 ) + 1 injective edge coloring graph, i.e., it is not a perfect EIC-graph. Proof. Since Kn is a subgraph of Kn2 , then χi (Kn2 ) ≥ n(n2+1 ) . Suppose that there exist a coloring c using n(n2+1 ) colors to such that Kn2 is an injective coloring. Without loss of generality, assume that c (e0 ) = i. Then there exist an edge colored by i in Kn , say e1 = wu. Similarly, there is an edge e1 (= w u ) ∈ E (Kn ) such that c (e1 ) = i. If e1 or e1 is not incident to v, then e0 , vw (or v w ) and e1 (or e1 ) can form three consecutive edges in Kn2 , a contradiction. Now, both e1 and e1 are adjacent to the edge e0 , then e1 , e0 and e1 form three consecutive edges in Kn2 , a contradiction. Then χi (Kn2 ) ≥ n(n2+1 ) + 1.
Now we give a coloring of Kn2 using n(n2+1 ) + 1 colors such that Kn2 is an injective edge coloring graph. The coloring is defined as follows. Firstly, we color each edge incidenting with v in Kn by a different color in [1, n], and color each edge incidenting with v in Kn by colors in [n + 1, 2n]. Then color the other edges in Kn by colors in [n + 1, n(n2+1 ) ], and color the other edges in Kn by colors [1, n] ∪ [2n + 1, n(n2+1 ) ]; then color the edge e0 by a new color. It is easy to check that Kn2 is an injective edge coloring graph. Then χi (Kn2 ) = n(n2+1 ) + 1.
Let F be a class of complete graphs of order at most t and Kt be a complete graph with vertex set {v1 , v2 , . . . , vt }. Let G0 be a graph obtained as follows. We coalesce some complete graphs in F with the vertex vi of Kt such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s , where ni be the maximum order of the complete graphs coalescing to vi and 0 ≤ i ≤ s ≤ t. Note that every vertex in Kt may be coalesced many different complete graphs. Lemma 4.1. The graph G0 is a perfect EIC-graph. Proof. Since Kt is a subgraph of G0 , then χi (G0 ) ≥ t (t 2−1 ) . If we can give an injective edge coloring of G0 using t (t 2−1 ) colors, the lemma can be proved. Let s0 be the maximum number such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s . Without loss generality, we assume n1 ≥ n2 ≥ · · · ≥ ns0 . We claim that all the complete graphs of order t only coalesced to the only vertex in Kt . If not, the graph Kt2 is a subgraph of G0 and n1 + n2 = 2(t − 1 ), but s(t − 1 ) − 2s = 2(t − 1 ) − 1, a contradiction.
Let c : V [G0 ] → [ t (t 2−1 ) ] be a t (t 2−1 ) -coloring of G0 . The coloring c is defined as follows. We firstly color each edge of Kt
with a different color in [ t (t 2−1 ) ]; and then color the every complete graph coalesced to vi ∈ V (Kt ), color the edge incidenting
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to vi in each complete graph with the colors in c[vi ] \ c[G0 [v1 , . . . , vi−1 ]], color the other edges not adjacent to vi using a different color in [ t (t 2−1 ) ] \ c[vi ]. For the structure of G0 , we know that the edges of every complete graph in F incidenting with different vertices in Kt must receive the different colors; otherwise, there exist three consecutive edges e1 , e2 , e3 such that c (e1 ) = c (e2 ), a contradiction. Similarly, the edges of the complete graphs in F incidenting with vi of Kt is colored by different colors with the edges not adjacent to vi in Kt . By some calculation and check the consecutive edges in G0 , it is easy to check c is an injective edge coloring of G0 . The lemma is obtained. Note that if the graphs in F are K2 , then the above lemma becomes the Proposition 2.1. Now, we give the following perfect EIC-graphs class, which is stated as the following theorem. Theorem 4.1. Let G be a block graph with the maximum clique of order t ( ≥ 3). And for each clique B of G, if V (B ) = {v1 , v2 , . . . , vs } and the blocks adjacent to the vertices of B in G such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s , where ni be the number of the maximum order of the complete graphs coalescing to vi , 0 ≤ i ≤ s ≤ t. This condition is called block-condition. Then G is a perfect EIC-graph. Proof. We argue this theorem by induction on |V(G)|. The conclusion holds trivially if |V (G )| = t. So we assume that this theorem holds for the small |V(G)|. Since G is a block graph, then there exist two end-blocks, say B0 and B1 . Without loose generality, assume that |B0 | ≤ |B1 |. In G, set the block adjacent to the block B0 in G be C0 and C1 , C2 , . . . , Cm be the blocks adjacent C0 in G, where 0 ≤ m ≤ t. Set G = G0 − B0 . If there exists an injective edge coloring of c : V (G ) → [ t (t 2−1 ) ] such that G is an injective edge coloring graph, then we want to extend the coloring c of G to be an injective edge coloring c of G. According to the size of B0 , the following cases are considered. Case 1 |V (B0 )| = t In this case, since |B0 | ≤ |B1 |, B1 is a clique of order t in G . So G is block graph with the maximum clique of order t, the block-condition is hold for each block of G . Then there exists an injective edge coloring c of G . Since the G is a block graph, then we only consider the subgraph G [C0 ∪ C1 ∪ ∪ Cm ]. Now the subgraph G [C0 ∪ C1 ∪ ∪ Cm ] ∪ B0 is the case of the Lemma 4.1. Then the result holds. Case 2 |V(B0 )| < t In this case, the only different is the order of B0 is least t. However, we can use the colors of [ t (t 2−1 ) ]. So it is easily to make G be an injective edge coloring c by extending the injective edge coloring c of G . Then the result holds. Note that Proposition 2.2 is a special case of this theorem, when the maximum clique is only K3 and all the other cliques are K2 . Acknowledgments Supported by the Joint Funds of Department of Education under the Natural Science Foundation of Shandong Province (Grant No. ZR2014JL001) of China and the Excellent Young Scholars Research Fund of Shandong Normal University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J.A. Bondy, U.S.R. Murty, Graph Theory, GTM 244, Springer, 2008. Y. Bu, D. Chen, A. Raspaud, W. Wang, Injective coloring of plane graphs, Discrete Appl. Math. 157 (2009) 663–672. Y. Bu, K. Lu, Injective coloring of plane graphs with girth 7, Discrete Math. Algorithms Appl. 4 (2012) 8. 1250034. D.M. Cardoso, J.O. Cerdeira, J.P. Cruz, C. Dominic, Injective edge chromatic index of a graph, http://arxiv.org/abs/1510.02626. W.D. Cranston, S.J. Kim, G. Yu, Injective coloring of graphs with low average degree, Algotithmica 60 (2011) 553–568. A. Doyon, G. Hahn, A. Raspaud, Some bounds on the injective chromatic number of graphs, Discrete Math. 310 (2010) 585–590. G. Hahn, J. Kratochvíl, J. Širánˇ , D. Sottean, On the injective chromatic number of graphs, Discrete Math. 256 (2002) 179–192. S. Li, X. Li, Y. Shi, Note on the complexity of deciding the rainbow (vertex-)connectedness for bipartite graphs, Appl. Math. Comput. 258 (2015) 155–161. X. Li, Y. Sun, Rainbow Connections of Graphs, Springer, New York, 2012. X. Li, Y. Shi, Y. Sun, Rainbow connections of graphs—a survey, Graphs & Combin. 29 (2013) 1–38. B. Lužar, R. Škrekovski, M. Tancer, Injective coloring of plane graphs with few colors, Discrete Math. 309 (2009) 5636–5649. M. Chen, G. Hahn, A. Raspaud, W. Wang, Some results on the injective chromatic number of graphs, J. Comb. Optim. 24 (2012) 299–318. R. Diestel, Graph Theory, GTM 173, Springer, 2006.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Note on the perfect EIC-graphs Jun Yue a, Shiliang Zhang b,∗, Xia Zhang a a b
School of Mathematical Sciences, Shandong Normal University, Jinan 250014, Shandong, China Mechanical and Electrical Engineering Department, Jinan Engineering Vocational Technical College, Jinan 250200, Shandong, China
a r t i c l e
i n f o
Keywords: Edge coloring Injective coloring Injective edge coloring Injective edge chromatic index Perfect EIC-graph
a b s t r a c t Three edges e1 , e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length 3. The injective edge coloring number χi (G ) is the minimum number of colors permitted in a coloring of the edges of G such that if e1 , e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. Let ω denote the number of edges in a maximum clique of G. A graph G is called an ω edge injective colorable (or perfect EIC-)graph if χi (G ) = ω . In this paper, we give a sharp bound of the injective coloring number of a 2-connected graph with some forbidden conditions, and then we also characterize some perfect EIC-graph classes, which extends the results of perfect EIC-graph of Cardoso et al. in [Injective edge chromatic index of a graph, http://arxiv.org/abs/1510.02626.]. © 2016 Elsevier Inc. All rights reserved.
1. Introduction All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty [1]. Except the classical vertex coloring and edge coloring, there are many kinds of colorings are studied, such as list coloring, star coloring and acyclic coloring. In addition, rainbow connection and rainbow vertex-connection are the new ones, we refer to a survey [10] and some new papers [8,9]. Recently, a new edge coloring–injective edge coloring, which has concrete applications in network science, causes researchers’ attentions. In this paper, we will focus on it. An injective coloring of G is a coloring of the vertices of G such that for every vertex v ∈ V (G ), all the neighbors of v are assigned distinct colors, i.e., if x and y are two distinct neighbors of v, then c(x) = c(y). The smallest integer k such that G has an injective k-coloring is the injective chromatic number of G, denoted by χ i (G). Injective coloring of graphs was introduced by Hahn et al. in [7] and was originated from Complexity Theory on Random Access Machines, and can be applied in the theory of error correcting codes [7]. In the same paper, they proved that, for k ≥ 3, it is NP-complete to decide whether the injective chromatic number of a graph is at most k. Since then, many researchers studied on this coloring and got many beautiful results. For more details, we refer to [2,3,5,6,11,12]. Similar to the injective coloring, an edge version of the injective coloring was introduced by Cardoso et al. in [4]. An injective edge coloring of a graph G is an edge coloring of G such that if e1 , e2 and e3 are consecutive edges in G, then e1 and e3 receive the different colors. The injective edge coloring number or injective edge chromatic index of a graph G, denoted by χi (G ), is the minimum number of colors permitted in an injective edge coloring of G. In [4], Cardoso et al. gave the exact values of injective edge coloring number for several classes of graphs, such as path, complete bipartite graph, complete
∗
Corresponding author. Tel.: +86 15966063925. E-mail addresses: [email protected] (J. Yue), [email protected] (S. Zhang), [email protected] (X. Zhang).
http://dx.doi.org/10.1016/j.amc.2016.05.031 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.
482
J. Yue et al. / Applied Mathematics and Computation 289 (2016) 481–485
Fig. 1. A block graph G and a friendship graph Fn .
graph and so on. And further, they also gave some bounds on injective edge coloring number of some graph and proved that checking whether χi (G ) = k is NP -complete. The clique number of a graph G, denoted by ω(G), is the number of vertices in a maximum clique of G. The number of edges in a maximum clique of G is denoted by ω (G). Obviously, ω (G ) = ω (G )(ω2(G )−1 ) . It is easy to see that χ i (G) ≥ ω(G) and χi (G ) ≥ ω (G ). We say that G is an ω edge injective colorable (perfect EIC-)graph if χi (G ) = ω . In [4], the authors gave some perfect EIC-graphs, such as the complete graph, the star, the friendship graph and the unicyclic graph with K3 . However, there are many unknown perfect EIC-graphs. In the same paper, the authors also gave the following open problem, which is stated as follows. Problem 1.1. How to characterize the perfect EIC-graph, i.e., how to use the structure of the graph to characterize the graph with χi (G ) = ω (G ). In this paper, we consider the injective edge coloring of graphs. We then determine injective edge colorings and injective edge coloring numbers. In the next section, we give some basic definitions and concepts, which will be used in the following sections. And then in Section 3, we give a sharp bound of a 2-connected graph by using ear decomposition, and also we construct a 2-connected perfect EIC-graph class. Section 4 gives the construction of a graph which is an perfect EIC-graph with cut-vertex. 2. Preliminaries In this section, we firstly start with some basic concepts and definitions. And then we list some known results, which will be used in the following sections. For other notation and terminology, we refer to [1] and [13]. Let G = (V, E ) be a graph. The number of vertices of a graph G is its order, written as n; its number of edges is its size, denoted by m. The degree d (v ) of a vertex v in G is the number of edges at v. If d (v ) = 1, the vertex v is called a pendant vertex. And the edge is called a pendant edge if one of its vertices is a pendant vertex. As usually, the path, the cycle, and the complete graph with n vertices will be denoted by Pn , Cn and Kn , respectively. A vertex v is called a cut-vertex in G if G − v contains more components than G does; in particular if G is connected, then a cut-vertex is a vertex v such that G − v is disconnected. Similarly, a bridge (or cutedge) is an edge whose deletion increases the number of components. A block in a graph G is a maximal nonseparable subgraph—that is, a nonseparable subgraph that is not properly contained in any other nonseparable subgraph of G. A nonseparable graph is itself often called a block. K2 is a block, but obviously no other block can contain a bridge. The clique is a subgraph of G such that it is a complete graph. The clique number ω(G) is the number of vertices in a maximum clique of G. The number of edges in a maximum clique of G is denoted by ω (G). If G has size m ≥ 1, then ω (G ) = ω (G )(2ω−1 ) . Three edges e1 , e2 and e3 in a graph G are consecutive if they form a path (in this order) or a cycle of length 3. A block graph (or clique tree) G is a graph in which every biconnected component (block) is a clique (see Fig. 1). A block of a graph G containing only one cut vertex is called an end block of G. A friendship graph Fn is a planar graph with 2n + 1 vertices and 3n edges, i.e., the graph with 2n + 1 vertices formed by n ≥ 1 triangles all attached to a common vertex (see Fig. 1). Now, we list some known results about the perfect EIC-graphs, which are stated as follows. Lemma 2.1 ([4]). For any connected graph G of order n ≥ 2, χi (G ) ≥ ω (G ). Example 2.1 ([4]). The following graphs are examples of perfect EIC-graphs. 1. The complete graph Kn ; 2. The start K1,q ; 3. The friendship graph Fn . Before giving the following proposition, we need a operation cornoa, which is stated as follows. The cornoa Kp ◦ K1 is the graph obtained from Kp by adding a pendant edge to each of its vertices.
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Proposition 2.1 ([4]). For any positive integer p ≥ 3, consider the complete graph Kp with V (K p ) = {v1 , . . . , v p }, and a family of stars K1,q1 , . . . , K1,q p , with qj ≥ 1. Let G be the graph obtained coalescing a maximum degree vertex of the star K1,q j with the vertex v j of the Kp , for j = 1, . . . , p. Then G is a perfect EIC-graph. Similarly to the graph of the above proposition, we present a construction of the perfect EIC-graph as follows. Lemma 2.2. For any positive integer n ≥ 1, consider the friendship graph Fn whose common vertex is v and triangles are labeled with T1 , T2 , . . . , Tn and a family of complete graphs Kq1 , Kq2 , . . . , Kqs with 0 ≤ s ≤ n. Let G be a graph obtained by replacing the triangle Ti in Fn by the complete graph Ki , where 1 ≤ i ≤ s. Then G is a perfect EIC-graph. Proof. Let qi and mi be the order and size of the complete graph Kqi , where 1 ≤ i ≤ s. Suppose that m1 ≤ m2 ≤ ≤ ms . By Lemma 2.1, we can get χi (G ) ≥ ms , since the complete graph Kqs is a subgraph of G . Now, we only to show that there exists a coloring c of G using ms colors to make G be an injective edge coloring graph. We color the edges of G as follows. For Kqs , color each edge incidenting with v with a different color in [1, ns ], and then color the other edges with different colors in [ns + 1, ms ]; for Kqi , color each edges incidenting with v by a different color in [1, ni ], and then color the other edges in Kqi with different colors in [ms + ni − mi , ms ]. It is easy to check that this coloring c is an injective edge coloring of G . The result is obtained. Note that if s = n and all the complete graphs are K2 , then G is a star K1,n ; if s = n and all the complete graphs are K3 , then G is the friendship graph. To end this section, we listed the following known result about the unicyclic graphs. Proposition 2.2 ([4]). If G is a unicyclic graph with K3 , then G is a perfect EIC-graph. 3. Injective edge coloring of 2-connected graph We start this section with a famous graph decomposition, named ear decomposition, of a 2-connected graph, which can be used to characterize several important classes of graphs. Let H be a subgraph of a graph G. An ear of H in G is a nontrivial path in G whose end vertices lie in H but whose internal vertices are not. An ear decomposition of a 2-connected graph G is a sequence G0 , G1 , . . . , Gk of 2-connected subgraph of G such that (1) G0 is a cycle of G; (2) Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of Gi−1 in G; (3) Gi−1 (1 ≤ i ≤ k) is a proper subgraph of Gi ; (4) Gk = G. Now, we give the following lemma which will be used in this section. Lemma 3.1. Let H be a graph with injective edge chromatic number χi (H ) ≥ (H ) + 1. If Pn = v0 , v1 , . . . , vn is an ear of H and v0 , vn ∈ V (H ), n ≥ 4 or n = 2, then χi (G ) ≤ χ (H ) + 1, where G = H ∪ Pn . Proof. Let c: E(H) → [k] be an injective edge coloring of H. Let c[v] be the set of colors of the edges incidenting with v ∈ V (H ). Since χi (Pn ) = 2, for n ≥ 4 and χi (H ) ≥ + 1, we can color the Pn using a new color k + 1 and a color k0 of c \ c[v0 ] and a color kn of c \ c[vn ]. Now, we color the graph G as follows. First, keep the injective edge coloring of H. And then color the edges v0 v1 , vn−1 vn with the color k + 1, and color the leave edges of Pn with k0 , kn . We can check that it is an injective edge coloring of G. Then χi (G ) ≤ χ (H ) + 1 holds. Similarly to the above lemma, we give the following observation. Observation 3.1. Let H be a graph with injective edge chromatic number χi (H ) ≥ (H ) + 1. If Pn = v0 , v1 , . . . , v3 is an ear of H and v0 , v3 ∈ V (H ), then χi (G ) ≤ χ (H ) + 2, where G = H ∪ Pn . Using the above results, we get a bound of injective edge coloring of a 2-connected graph G, which can be stated as follows. Theorem 3.1. Let G be a 2-connected graph. And G0 , G1 , . . . , Gk be an ear decomposition of G such that Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of length at least 2 of Gi−1 in G, then
χi (G ) =
2k + 2
if |G0 | = 0 (mod 4 ),
2k + 3
otherwise.
Furthermore, if Pi−1 is an ear of length at least 2 of Gi−1 in G and χi (Gi ) ≥ (Gi ) + 1, the number of ears of length 3 is t, then χi (G ) = k + t + 3. And the bounds are sharp. Proof. The proof of the theorem is easily got from Lemma 3.1 and Observation 3.1, since χi (Pn ) = 2, for n ≥ 4. Now we give the examples to show the bounds are sharp. The graph G is obtained from a cycle C2k+1 with an ear of length of t ≥ 4. It is easily to check that the injective edge coloring of G must use χi (C2k+1 ) + 1 colors. Then the theorem holds. Now, we construct a graph G∗ as follows. Let G0 = Kn where n ≥ 5; and Gi = Gi−1 ∪ Pi−1 , where Pi−1 is an ear of Gi−1 in G, and the degree of the vertices except the vertices in Kn is at most n2 for every Gi , where 1 ≤ i ≤ k; then G∗ = Gk .
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Fig. 2. The graph Kn2 .
Theorem 3.2. The graph G∗ is a perfect EIC-graph. Proof. Because the degree of the vertices except V(Kn ) is at most G∗ ,
then χi (G∗ ) ≥ n(n2−1 ) .
n 2,
Kn is the maximum clique of G∗ . Since Kn is an included
graph of Now, we give an injective edge coloring c of G∗ as follows. First, we color each edge of G0 with a different color in n (n−1 ) [ 2 ]. For 1 ≤ i ≤ k, color the ear Pi of Gi by the following cases. If two end vertices of Pi are in G0 , i.e., Pi = v0 , v1 , . . . , vs and v0 , vs ∈ V (G0 ), then we color the ear Pi using two colors which are colored the edges v01 vs1 , v02 vs2 , where v0i ∈ NG0 (v0 ) and vsi ∈ NG0 (vs ), i ∈ {1, 2}. If two end vertices of Pi = v0 , v1 , . . . , vs are not in G0 , then there must exist a color k0 that not used by the edges which can be form the consecutive edges with v0 v1 in Gi , since d (v0 ) ≤ 2n . Similarly, there also exists a color ks that is not used by the edges which can be form the consecutive edges with vs−1 vs in Gi . Note that d (v0 ) and d (vs ) ≤ n2 , then k0 = ks . If on end vertex of Pi is in G0 and another is not in G0 , i.e., v0 ∈ V (G0 ). Since d (vs ) ≤ n2 , then there must be two different colors to color the ear Pi such that Gi an injective edge coloring graph. Then we can check that the coloring c of G∗ an injective edge coloring. Then the theorem holds. 4. Perfect EIC-graph with cut-vertex In this section, we first give some basic lemmas, which will be used in the following section. And then we construct a class of ω EIC-graphs, which extends the results in [4]. Now, we begin this section with a special graph which is not a perfect EIC-graph. Let Kn2 be a graph obtained by connecting Kn and Kn with an edge e0 = vv , where v ∈ V (Kn ), v ∈ V (Kn ), (see Fig. 2). Observation 4.1. Kn2 is a n(n2−1 ) + 1 injective edge coloring graph, i.e., it is not a perfect EIC-graph. Proof. Since Kn is a subgraph of Kn2 , then χi (Kn2 ) ≥ n(n2+1 ) . Suppose that there exist a coloring c using n(n2+1 ) colors to such that Kn2 is an injective coloring. Without loss of generality, assume that c (e0 ) = i. Then there exist an edge colored by i in Kn , say e1 = wu. Similarly, there is an edge e1 (= w u ) ∈ E (Kn ) such that c (e1 ) = i. If e1 or e1 is not incident to v, then e0 , vw (or v w ) and e1 (or e1 ) can form three consecutive edges in Kn2 , a contradiction. Now, both e1 and e1 are adjacent to the edge e0 , then e1 , e0 and e1 form three consecutive edges in Kn2 , a contradiction. Then χi (Kn2 ) ≥ n(n2+1 ) + 1.
Now we give a coloring of Kn2 using n(n2+1 ) + 1 colors such that Kn2 is an injective edge coloring graph. The coloring is defined as follows. Firstly, we color each edge incidenting with v in Kn by a different color in [1, n], and color each edge incidenting with v in Kn by colors in [n + 1, 2n]. Then color the other edges in Kn by colors in [n + 1, n(n2+1 ) ], and color the other edges in Kn by colors [1, n] ∪ [2n + 1, n(n2+1 ) ]; then color the edge e0 by a new color. It is easy to check that Kn2 is an injective edge coloring graph. Then χi (Kn2 ) = n(n2+1 ) + 1.
Let F be a class of complete graphs of order at most t and Kt be a complete graph with vertex set {v1 , v2 , . . . , vt }. Let G0 be a graph obtained as follows. We coalesce some complete graphs in F with the vertex vi of Kt such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s , where ni be the maximum order of the complete graphs coalescing to vi and 0 ≤ i ≤ s ≤ t. Note that every vertex in Kt may be coalesced many different complete graphs. Lemma 4.1. The graph G0 is a perfect EIC-graph. Proof. Since Kt is a subgraph of G0 , then χi (G0 ) ≥ t (t 2−1 ) . If we can give an injective edge coloring of G0 using t (t 2−1 ) colors, the lemma can be proved. Let s0 be the maximum number such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s . Without loss generality, we assume n1 ≥ n2 ≥ · · · ≥ ns0 . We claim that all the complete graphs of order t only coalesced to the only vertex in Kt . If not, the graph Kt2 is a subgraph of G0 and n1 + n2 = 2(t − 1 ), but s(t − 1 ) − 2s = 2(t − 1 ) − 1, a contradiction.
Let c : V [G0 ] → [ t (t 2−1 ) ] be a t (t 2−1 ) -coloring of G0 . The coloring c is defined as follows. We firstly color each edge of Kt
with a different color in [ t (t 2−1 ) ]; and then color the every complete graph coalesced to vi ∈ V (Kt ), color the edge incidenting
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to vi in each complete graph with the colors in c[vi ] \ c[G0 [v1 , . . . , vi−1 ]], color the other edges not adjacent to vi using a different color in [ t (t 2−1 ) ] \ c[vi ]. For the structure of G0 , we know that the edges of every complete graph in F incidenting with different vertices in Kt must receive the different colors; otherwise, there exist three consecutive edges e1 , e2 , e3 such that c (e1 ) = c (e2 ), a contradiction. Similarly, the edges of the complete graphs in F incidenting with vi of Kt is colored by different colors with the edges not adjacent to vi in Kt . By some calculation and check the consecutive edges in G0 , it is easy to check c is an injective edge coloring of G0 . The lemma is obtained. Note that if the graphs in F are K2 , then the above lemma becomes the Proposition 2.1. Now, we give the following perfect EIC-graphs class, which is stated as the following theorem. Theorem 4.1. Let G be a block graph with the maximum clique of order t ( ≥ 3). And for each clique B of G, if V (B ) = {v1 , v2 , . . . , vs } and the blocks adjacent to the vertices of B in G such that n1 + n2 + · · · + ns ≤ s(t − 1 ) − 2s , where ni be the number of the maximum order of the complete graphs coalescing to vi , 0 ≤ i ≤ s ≤ t. This condition is called block-condition. Then G is a perfect EIC-graph. Proof. We argue this theorem by induction on |V(G)|. The conclusion holds trivially if |V (G )| = t. So we assume that this theorem holds for the small |V(G)|. Since G is a block graph, then there exist two end-blocks, say B0 and B1 . Without loose generality, assume that |B0 | ≤ |B1 |. In G, set the block adjacent to the block B0 in G be C0 and C1 , C2 , . . . , Cm be the blocks adjacent C0 in G, where 0 ≤ m ≤ t. Set G = G0 − B0 . If there exists an injective edge coloring of c : V (G ) → [ t (t 2−1 ) ] such that G is an injective edge coloring graph, then we want to extend the coloring c of G to be an injective edge coloring c of G. According to the size of B0 , the following cases are considered. Case 1 |V (B0 )| = t In this case, since |B0 | ≤ |B1 |, B1 is a clique of order t in G . So G is block graph with the maximum clique of order t, the block-condition is hold for each block of G . Then there exists an injective edge coloring c of G . Since the G is a block graph, then we only consider the subgraph G [C0 ∪ C1 ∪ ∪ Cm ]. Now the subgraph G [C0 ∪ C1 ∪ ∪ Cm ] ∪ B0 is the case of the Lemma 4.1. Then the result holds. Case 2 |V(B0 )| < t In this case, the only different is the order of B0 is least t. However, we can use the colors of [ t (t 2−1 ) ]. So it is easily to make G be an injective edge coloring c by extending the injective edge coloring c of G . Then the result holds. Note that Proposition 2.2 is a special case of this theorem, when the maximum clique is only K3 and all the other cliques are K2 . Acknowledgments Supported by the Joint Funds of Department of Education under the Natural Science Foundation of Shandong Province (Grant No. ZR2014JL001) of China and the Excellent Young Scholars Research Fund of Shandong Normal University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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