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Acyclic vertex coloring of graphs of maximum degree six
Discrete Mathematics 325 (2014) 17–22
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Acyclic vertex coloring of graphs of maximum degree six Yancai Zhao a , Lianying Miao b,∗ , Shiyou Pang b , Wenyao Song b a
Wuxi City College of Vocational Technology, Wuxi, Jiangsu, 221116, PR China
b
School of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, PR China
article
info
Article history: Received 31 July 2013 Received in revised form 22 January 2014 Accepted 27 January 2014 Available online 4 March 2014
abstract In this paper, we prove that every graph with maximum degree six is acyclically 10colorable, thus improving the main result of Hervé Hocquard (2011). © 2014 Published by Elsevier B.V.
Keywords: Graph coloring Bounded degree graphs Acyclic coloring
1. Introduction A proper vertex coloring of a graph G = (V , E ) is an assignment of colors to the vertices of G such that two adjacent vertices do not use the same color. A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles; in other words, the graph induced by every two color classes is a forest. The acyclic chromatic number of G, denoted by χa (G), is the smallest integer k such that G is acyclically k-colorable. Acyclic colorings were introduced by Grünbaum [10]. The following are some results about acyclic colorings of graphs. Theorem 1.1 ([10]). Every planar graph is acyclically 9-colorable. Theorem 1.2 ([4]). Every planar graph is acyclically 5-colorable. This bound is tight since there exist 4-regular planar graphs [10] which are not acyclically 4-colorable. Theorem 1.3 ([2]). Every graph with maximum degree ∆ can be acyclically colored using O(∆(G)4/3 ) colors. Theorem 1.4 ([1]). Every graph with maximum degree ∆ can be acyclically colored using ∆(∆ − 1) + 2 colors. For graphs with maximum degree six, there are the following results. Theorem 1.5 ([18]). Every graph of maximum degree 6 can be acyclically colored with 12 colors. Theorem 1.6 ([11]). Every graph of maximum degree 6 can be acyclically colored with 11 colors. Other results about the acyclic coloring of graphs can be seen in [1,5,8,6,7,9–16,19]. Here we improve Theorem 1.6 by proving that. Theorem 1.7. Every graph with maximum degree six is acyclically 10-colorable.
∗
Corresponding author. E-mail addresses: [email protected] (Y. Zhao), [email protected] (L. Miao).
http://dx.doi.org/10.1016/j.disc.2014.01.022 0012-365X/© 2014 Published by Elsevier B.V.
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Y. Zhao et al. / Discrete Mathematics 325 (2014) 17–22
Fig. 1. An illustration of NC (u), nC (u), Cϕ (u) and cϕ (u).
This theorem also answers the second question posed by Hervé Hocquard [11]. We now introduce the notations (some of them are first given in [11]) and use the standard graph theory terminology [17] not defined here. Let G = (V (G), E (G)), and v ∈ V (G). We use N (v) and d(v) to denote the set of the neighbors and the degree of v in G respectively. A partial acyclic coloring ϕ of G is an assignment of colors to a subset U of V (G) such that ϕ is an acyclic coloring of G[U ]. Let ϕ be a partial acyclic coloring of G with the color set C and the colored subset U ⊆ V (G) and let v be an uncolored vertex of G. We say that a color c ∈ C is available for v if no neighbor of v is colored c. We say that a color c ∈ C is feasible for v if it is available for v and coloring v with c results in a partial acyclic coloring of G. We say that a color c ∈ C is no-feasible for v if it is available for v and coloring v with c results in bicolored cycles in G. Let Fv and NFv denote the set of feasible and no-feasible colors for v . For a vertex u ∈ V (G) (colored or uncolored), we denote the set and the number of colored neighbors of u by NC (u) = N (u) ∩ U and nC (u) = |NC (u)| respectively. We denote by Cϕ (u) the set of colors used by vertices in NC (u) and cϕ (u) = |Cϕ (u)|. For example, in Fig. 1, NC (u) = {v1 , v2 , v3 , v4 , v5 }, nC (u) = 5, Cϕ (u) = {1, 2, 3, 4} and cϕ (u) = 4. Finally, we denote by ∆(G), the maximum degree of a graph G. We assume that the graphs in this paper are connected. Let C = {1, 2, . . . , 10}. 2. Main result It is known that [3, P34] every graph of maximum degree at most ∆ is an induced subgraph of a ∆-regular graph, and it is sufficient to consider 6-regular connected graphs in this paper. The following definition is first given in [11]. Let G be a ∆-regular connected graph. A good spanning tree of G is a spanning tree T such that T contains a vertex adjacent to ∆ − 1 leaves. Lemma 1 ([11]). Every regular connected graph admits a good spanning tree. Remark 1. The idea of the proof of Theorem 1.7 is mainly from [11]. We make more careful analysis and use one new technique of constructing bipartite graphs to reduce the number of colors needed to 10. Proof of Theorem 1.7. Let G be a 6-regular connected graph. Let T be a good spanning tree of G. Let xn be a vertex adjacent to five leaves x1 , x2 , x3 , x4 , x5 in T . We order the vertices of G from x1 to xn according to a post-order walk of T . First, we color x1 , x2 , x3 , x4 , x5 with five distinct colors. Then we will successively color x6 , x7 , . . . , xn−1 while the colors of x1 , x2 , x3 , x4 , x5 will never be changed. Finally, we color xn . Suppose that we have colored x1 , x2 , . . . , xi−1 (6 ≤ i ≤ n − 1). Let ϕ be an acyclic 10-coloring of Gi−1 = G[x1 , x2 , . . . , xi−1 ]. Now we color xi = u. Since u is adjacent to at least one of xi+1 , . . . , xn , we have nC (u) ≤ 5. W.l.o.g. j assume that nC (u) = 5. Let NC (u) = {v1 , v2 , v3 , v4 , v5 }. For 1 ≤ i, j, k ≤ 5, let N (vi )\{u} = {vi |1 ≤ j ≤ 5} and j
j ,k
N (vi )\{vi } = {vi |1 ≤ k ≤ 5}. Let A = {x1 , x2 , x3 , x4 , x5 }. (See Fig. 2.) Since u ̸= xn , we have the following claim.
Claim 1. If v ∈ N (u) and nC (v) = 5, then v ̸∈ A. If v ̸∈ N (u) and nC (v) = 6, then v ̸∈ A. Construction. We construct a bipartite graph H with the bipartition (X , Y ) such that X = {x|x ∈ NC (u) and there is a vertex x′ ∈ NC (u) such that ϕ(x) = ϕ(x′ ) and x ̸= x′ } and Y = NFu . For any x ∈ X and y ∈ Y , x is adjacent to y in H iff assigning u the color y will result in a bicolored cycle passing through u and x. It is easy to see that dH (y) ≥ 2 for any y ∈ Y if X ̸= φ and Y ̸= φ. Now we consider the following five cases. Case 1. cϕ (u) = 5. Then there remain five colors for u. Case 2. cϕ (u) = 4. W.l.o.g. assume that ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = 2, ϕ(v4 ) = 3, ϕ(v5 ) = 4. We can color u with a color in C \{Cϕ (u) ∪ [Cϕ (v1 ) ∩ Cϕ (v2 )]} (which is not φ ).
Y. Zhao et al. / Discrete Mathematics 325 (2014) 17–22
j
19
j,k
Fig. 2. An illustration of vi , vi , vi .
Case 3. cϕ (u) = 3. Then exactly one color or two colors are repeated among the neighbors of u. If it is the former, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = 2, ϕ(v5 ) = 3. Otherwise, we can assume that ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = ϕ(v4 ) = 2, ϕ(v5 ) = 3. Case 3.1. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = 2, ϕ(v5 ) = 3. If Fu ̸= φ , we are done. Otherwise NFu = C \{ 1, 2, 3}. In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 } and Y = NFu . Since y∈Y dH (y) ≥ 14, at least one of v1 , v2 , v3 , say v1 , has dH (v1 ) = 5, that is cϕ (v1 ) = 5. By Claim 1 we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. Case 3.2. ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = ϕ(v4 ) = 2, ϕ(v5 ) = 3. If Fu ̸= φ , we are done. Assume that Fu = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 } and Y = C \{1, 2, 3}. If one of v1 , v2 , v3 , v4 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 4, and then dH (vi ) ≤ 4 for all 1 ≤ i ≤ 4. Since y∈Y dH (y) ≥ 14 and dH (v1 ) = dH (v2 ), dH (v3 ) = dH (v4 ), so either dH (v1 ) = dH (v2 ) = 4 or dH (v3 ) = dH (v4 ) = 4, say dH (v1 ) = dH (v2 ) = 4. We can assume that v1 ̸∈ A. If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. Assume nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 4, ϕ(v13 ) = 5, ϕ(v14 ) = 6, ϕ(v15 ) = 7. Then {8, 9, 10} ⊆ Cϕ (v3 ) ∩ Cϕ (v4 ) since Fu = φ . If one of 2, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), we can recolor v1 with this color, then it becomes Case 2 or Case 3.1. Assume {2, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since dH (v1 ) = 4, 1 is repeated among the neighbors of v11 or v12 , by Claim 1 we can recolor v11 or v12 with 3 and recolor v1 with 8, and then it becomes Case 2. Case 4. cϕ (u) = 2. Then exactly one color or two colors are repeated among the neighbors of u. If it is the former, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ) = 1, ϕ(v5 ) = 2. Otherwise, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = ϕ(v5 ) = 2. Case 4.1. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ) = 1, ϕ(v5 ) = 2. If one of v1 , v2 , v3 , v4 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), and then it becomes Case 3. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 4. If F ̸= φ , we are done. Assume that F = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 } and Y = C \{1, 2}. Clearly y∈Y dH (y) ≥ 16 and dH (vi ) ≤ 4 for all 1 ≤ i ≤ 4. Thus dH (x) = 4 for any x ∈ X and dH (y) = 2 for any y ∈ Y . Since ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ), at least one of vi′ s(1 ≤ i ≤ 4) is not in A, say v1 . If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), then it becomes Case 3. Assume that nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 3, ϕ(v13 ) = 4, ϕ(v14 ) = 5 and ϕ(v15 ) = 6. If one of 7, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), say 7, we can recolor v1 with 7, then it becomes Case 3. Suppose {7, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since 3 ̸∈ Fu and dH (v1 ) = 4, there will be a bicolored cycle colored 1 and 3 passing through u, v1 and v11 or v12 if we color u with 3, so at least one of v11 and v12 has a neighbor (other than v1 ) colored 1, say v11 . By Claim 1 we can recolor v11 with 2 and recolor v1 with 7, and then it becomes Case 3. Case 4.2. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = ϕ(v5 ) = 2. If one of v1 , v2 , v3 , v4 , v5 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), and then it becomes Case 3. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 5. If Fu ̸= φ , we are done. Assume that Fu = φ . In this case, graph H in the Construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 , v5 } and Y = C \{1, 2}. Note that dH (vi ) ≤ 4 for all 1 ≤ i ≤ 5.
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Fig. 3. An illustration of above assumption, ◦ uncolored, ⊗ colored or uncolored.
Suppose one of v1 , v2 , v3 , v4 , v5 , say v1 , has dH (v1 ) = 4 and v1 ̸∈ A. If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), then it becomes Case 3. Assume that nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 3, ϕ(v13 ) = 4, ϕ(v14 ) = 5 and ϕ(v15 ) = 6. If one of 7, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), we can recolor v1 with this color, then it becomes Case 3. Suppose {7, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since 3 ̸∈ Fu and dH (v1 ) = 4, there will be a bicolored cycle colored 1 and 3 passing through u, v1 and v11 or v12 if we color u with 3, so there is one neighbor (other than v1 ) of v11 or v12 that colored 1, say v11 . Then by Claim 1 we can recolor v11 with 2 and recolor v1 with 7, and then it becomes Case 3. So assume that for all 1 ≤ i ≤ 5 either dH (vi ) ≤ 3 or dH (vi ) = 4 and vi ∈ A. Note that y∈Y dH (y) ≥ 16, dH (v4 ) = dH (v5 ) and at least one of v4 and v5 is not in A. We now assume that dH (v1 ) = 4, v1 ∈ A and dH (v2 ) = dH (v3 ) = 3, dH (v4 ) = dH (v5 ) = 3. Then y∈Y dH (y) = 16 and dH (y) = 2 for any y ∈ Y . We can assume that NH (v1 ) = {3, 4, 5, 6}, NH (v2 ) = {3, 4, 7}, NH (v3 ) = {5, 6, 7}, NH (v4 ) = NH (v5 ) = {8, 9, 10} and v4 ̸∈ A. Now we first consider v4 . If nC (v4 ) = 3, we can recolor v4 with 7, then it becomes Case 3. If nC (v4 ) = 4, we consider two cases: if cϕ (v4 ) = 4, we can recolor v4 with a color different from 1, 2 and those in Cϕ (v4 ), then it becomes Case 3. If cϕ (v4 ) = 3, then we may assume that ϕ(v41 ) = ϕ(v42 ) = 8. Then at least one of 1, 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ). We can recolor v4 with this color, and then it becomes Case 3 or Case 4.1. If nC (v4 ) = 5, we consider two cases: Case 4.2.I. cϕ (v4 ) = 4. W.l.o.g. assume that ϕ(v41 ) = ϕ(v42 ) = a. First suppose a ∈ {1, 3, 4, 5, 6, 7}, say a = 1 (the other cases are similar). If one of 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ). We can recolor v4 with this color, then it becomes Case 3. If {3, 4, 5, 6, 7} ⊆ Cϕ (v41 ) ∩ Cϕ (v42 ), by Claim 1 we can recolor v41 with 2 and recolor v4 with 3, then it becomes Case 3. Now suppose a ∈ {8, 9, 10}, say a = 8. W.l.o.g. assume that Cϕ (v4 ) = {1, 8, 9, 10}. Since 8 ̸∈ Fu and dH (v4 ) = 3, there will be a bicolored cycle colored 2 and 8 passing through u, v4 and v41 or v42 if we color u with 8, so there is one neighbor (other than v4 ) of v41 or v42 that colored 2, then at least one of 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ), say 3. We recolor v4 with 3, then it becomes Case 3. Case 4.2.II. cϕ (v4 ) = 3, and then Cϕ (v4 ) = {8, 9, 10}. We have two cases to consider: exactly one color or two colors are repeated among the neighbors of v4 . Case 4.2.II-1 It is the former case. We can assume that ϕ(v41 ) = ϕ(v42 ) = ϕ(v43 ) = 8, ϕ(v44 ) = 9, ϕ(v45 ) = 10. Since dH (v4 ) = 3, there is one neighbor (other than v1 ) of v41 or v42 or v43 that colored 2, say v41 . Let Y1 = {1, 3, 4, 5, 6, 7}. We assume that recoloring v4 with any color in Y1 will result in a bicolored cycle, for otherwise it will become C4.1 or C3 after recoloring v4 with the color. Now similarly to H, we construct a bipartite graph H1 with the bipartition (X1 , Y1 ) such that X1 = {v41 , v42 , v43 }. For any x ∈ X1 and y ∈ Y1 , x is adjacent to y in H1 iff reassigning v4 the color y will result in a bicolored cycle passing through v4 and x. We know that dH1 (y) ≥ 2 for any y ∈ Y1 . If dH1 (x) = 5 for some x ∈ X1 , we recolor x with the color in {1, 3, 4, 5, 6, 7} \ Cϕ (x), then it becomes Case 4.2.I. So assume dH1 (x) ≤ 4 for any x ∈ X1 . By the size of H1 , dH1 (x) = 4 for any x ∈ X1 . W.l.o.g. assume that NH1 (v41 ) = {1, 3, 4, 5}. Note that v41 has two neighbors colored 2, so v41 ̸∈ A. By Claim 1 we can recolor v41 with 6 to get Case 4.2.I. Case 4.2.II-2 It is the latter case. W.l.o.g. assume that ϕ(v41 ) = ϕ(v42 ) = 8, ϕ(v43 ) = ϕ(v44 ) = 9, ϕ(v45 ) = 10. Note that dH (v4 ) = 3, one of v41 and v42 passes through the bicolored cycle colored 2 and 8 if we color u with 8, say v41 , that is v41 has a neighbor (other than v1 ) colored 2. Similarly, one of v43 and v44 passes through the bicolored cycle colored 2 and 9 if we color u with 9, say v43 , that is v43 has a neighbor (other than v1 ) colored 2. We assume that recoloring v4 with any color in Y1 will result in a bicolored cycle, for otherwise it will become C4.1 or C3 after recoloring v4 with the color. Now we construct a bipartite graph H2 with the bipartition (X2 , Y1 ) such that X2 = {v41 , v42 , v43 , v44 }. For any x ∈ X2 and y ∈ Y1 , x is adjacent to y in H2 iff reassigning v4 color y will result in a bicolored cycle passing through v4 and x. (See Fig. 3.) We know that dH2 (y) ≥ 2 for any y ∈ Y1 . If dH2 (x) = 5 for some x ∈ X2 , we recolor x with the color in {1, 3, 4, 5, 6, 7} \ Cϕ (x), then it becomes Case 4.2.I. So assume dH2 (x) ≤ 4 for any x ∈ X2 . If dH2 (v41 ) = 4 or dH2 (v43 ) = 4, say v41 . By Claim we can recolor v41 with a color not in Cϕ (v4 ) ∪ Cϕ (v41 ) to get Case 4.2.I. So assume dH2 (x) = 3 for any x ∈ X2 .
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Fig. 4. An illustration of above assumption, ◦ uncolored, ⊗ colored or uncolored.
Then dH2 (y) = 2 for any y ∈ Y2 . We assume that NH2 (v41 ) = NH2 (v42 ) = {1, 3, 4} and NH2 (v43 ) = NH2 (v44 ) = {5, 6, 7}. (See Fig. 4). If both v41 and v43 are in A, we consider v2 . If nC (v2 ) = 3, we can recolor v2 with 5, then it becomes Case 3. If nC (v2 ) = 4, we consider two cases: if cϕ (v2 ) = 4, we can recolor v2 with a color different from 1, 2 and those in Cϕ (v2 ), then it becomes Case 3. If cϕ (v2 ) = 3, then we may assume that ϕ(v21 ) = ϕ(v22 ) = 3. If one of 5, 6, 8, 9 and 10, say 5, is not in Cϕ (v21 ) ∩ Cϕ (v22 ). We can recolor v2 with 5, and then it becomes Case 3. If {5, 6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), by Claim 1 we can recolor v21 , v22 with 1 and recolor v2 with 3, then it becomes Case 3. If nC (v2 ) = 5, we consider two cases. Case 4.2.I′ . cϕ (v2 ) = 4. W.l.o.g. assume that ϕ(v21 ) = ϕ(v22 ) = a. We first consider the case that a = 2. If {5, 6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), we recolor v21 with 1, recolor v2 with 5, then it becomes Case 3. If {5, 6, 8, 9, 10} ̸⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), say 5 ̸∈ Cϕ (v21 ) ∩ Cϕ (v22 ), we recolor v2 with 5, then it becomes Case 3. If a ∈ {3, 4, 7}, w.l.o.g. we assume that a = 3. Since dH (v2 ) = 3 and NH (v2 ) = {3, 4, 7}, there will be a bicolored cycle colored 1 and 3 passing through u, v2 and v21 or v22 if we color u with 3, so there is one neighbor (other than v2 ) of v21 or v22 that colored 1, say v21 . If C \ {2} ̸= Cϕ (v2 ) ∪ Cϕ (v21 ), choose a color in [C \ {2}] \ [Cϕ (v2 ) ∪ Cϕ (v21 )] to color v2 , then it becomes Case 3. If C \ {2} = Cϕ (v2 ) ∪ Cϕ (v21 ), we can recolor v21 with 2 and recolor v2 with a color different from 1, 2 and those in ϕ(v2 ), then it becomes Case 3. If a ∈ {5, 6, 8, 9, 10}, say a = 5, then {6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), for otherwise, we can recolor v2 with the color in {6, 8, 9, 10} \ [Cϕ (v21 ) ∩ Cϕ (v22 )], i,1
i,2
i,3
i,4
then it becomes Case 3. Let ϕ(v2 ) = 6, ϕ(v2 ) = 8, ϕ(v2 ) = 9, ϕ(v2 ) = 10 for i = 1, 2. If one of nC (v21 ) and nC (v22 ) is 6,
)∈ ̸ {6, 8, 9, 10}, we recolor v with a color α that is not in {6, 8, 9, 10, ϕ(v21,5 ), 3, 4, 5, 7} 1,5 1,5 and recolor v2 with a color not in {1, 2, 3, 4, 5, 7, α}, then it becomes Case 3. If ϕ(v2 ) ∈ {6, 8, 9, 10}, say v2 = 6. Now 1,1 1,5 1 ,1 1 ,5 1 1 we consider v2 , v2 and v2 . If 1 or 2 is not in ϕ(v2 ) ∩ ϕ(v2 ), we recolor v2 with 1 or 2 and recolor v2 with 6, then it 1,1 1 ,5 1 ,1 1,5 becomes Case 3. So we assume that {1, 2} ⊆ ϕ(v2 ) ∩ ϕ(v2 ). If one of 3, 4 and 7 is not in ϕ(v2 ) ∩ ϕ(v2 ), we recolor v21 1,1 1 ,5 1 ,1 with 3 this color, then it becomes the above case. So we assume that {3, 4, 7} ⊆ ϕ(v2 ) ∩ ϕ(v2 ). Then we recolor v2 and v21,5 with 5, and recolor v21 with 6, and recolor v2 with 8, and then it becomes Case 3. If nC (v21 ) = nC (v22 ) = 5. We assume that v21 ̸∈ A. We recolor v21 with 1 and recolor v2 with 6, and then it becomes Case 3. we assume nC (v ) = 6. If ϕ(v 1 2
1 ,5 2
1 2
Case 4.2.II′ . cϕ (v2 ) = 3. Similar to the above discussion in Case 4.2.II, either there are two neighbors of v2 , say v21 and v23 , which are in A or at least one of v21 and v23 is not in A. If both v21 and v23 are in A. We consider v3 . At the step that exactly two colors are repeated among the neighbors of v3 , all neighbors of v3 are not in A. So we can assume that one of v41 and v43 is not in A, say v41 . From now on we consider v41 . If the last neighbor of v41 is uncolored or colored a ̸∈ {1, 2, 3, 4}, we recolor v41 with one color different from those in Cϕ (v41 ) and those in Cϕ (v4 ). We obtain Case 4.2.I. Assume that the last neighbor of v41 is colored a ∈ {1, 2, 3, 4}. If the another neighbor of v43 is colored b ̸∈ {2, 5, 6, 7}, by Claim 1 we recolor v43 with one color different from those in Cϕ (v43 ) and those in Cϕ (v4 ), we obtain Case 4.2.I. Assume that the other neighbor of v43 is colored b ∈ {2, 5, 6, 7} or uncolored. 1 ,1
1,2
1 ,3
1,4
1 ,5
If a = 2, w.l.o.g. suppose ϕ(v4 ) = ϕ(v4 ) = 2, ϕ(v4 ) = 1, ϕ(v4 ) = 3 and ϕ(v4 ) = 4. Since dH (v4 ) = 3 and v41 1 ,1
passes through a bicolored cycle colored 2 and 8 after coloring u with 8, at least one of v4 than v
1 4)
1,1 4 .
colored 8, say v
1,2
and v4
has a neighbor (other
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1 ,2
If one of 5, 6, 7 and 9 is not in Cϕ (v4 ) ∩ Cϕ (v4 ), we recolor v41 with this color. We obtain Case 4.2.I. If {5, 6, 7, 9} ⊆ 1 ,1 4
1,2 4
1,1 4
1,1
Cϕ (v ) ∩ Cϕ (v ), then Cϕ (v ) = {5, 6, 7, 8, 9}. By Claim 1 we recolor v4 with 10, recolor v41 with 5 and recolor v4 with 3. Then it becomes Case 3. 1,1 1,2 1,3 1,4 1,5 If a ∈ {1, 3, 4}, say a = 1. W.l.o.g. suppose ϕ(v4 ) = ϕ(v4 ) = 1, ϕ(v4 ) = 2, ϕ(v4 ) = 3 and ϕ(v4 ) = 4. 1 ,1
Since NH2 (v41 ) = {1, 3, 4}, at least one of v4 1 ,1 4
7 and 9 is not in Cϕ (v 1,1 4
{5, 6, 7, 9} ⊆ Cϕ (v
1,2 4
1,2
and v4
1,1
has a neighbor (other than v41 ) colored 8, say v4 . If one of 5, 6,
) ∩ Cϕ (v ), we recolor v41 with this color, then it becomes Case 4.2.I or Case 4.2.II-1. Assume ) ∩ Cϕ (v41,2 ). By Claim 1 we recolor v41,1 with 10, recolor v41 with 5 and recolor v4 with 3. Then it
becomes Case 3. Case 5. cϕ (u) = 1. W.l.o.g. we assume that ϕ(vi ) = 1 for 1 ≤ i ≤ 5. If one of v1 , v2 , v3 , v4 , v5 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1 and those in Cϕ (v1 ), and then it becomes Case 4.1. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 5. If F ̸= φ , we are done. Assume that F = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 , v5 } and Y = C \{1}. Note that y∈Y dH (y) ≥ 18. We know that at least three of vi′ s(1 ≤ i ≤ 5) have dH (vi ) = 4, say v1 , v2 , v3 . W.l.o.g. we assume that v1 ̸∈ A. If nC (v1 ) = 4, then we can recolor v1 with a color different from 1 and those in Cϕ (v1 ), and then it becomes Case 4.1. So assume that nC (v1 ) = 5. W.l.o.g. we assume that ϕ(v11 ) = ϕ(v12 ) = 2, ϕ(v13 ) = 3, ϕ(v14 ) = 4, ϕ(v15 ) = 5. Since dH (v1 ) = 4, so at least one of v11 and v12 has a neighbor (other than v1 ) colored 1, so at least one of 6, 7, 8, 9 and 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ). We recolor v1 with this color, and then it becomes Case 4.1. Now we color xn . We need to consider two cases. (1) cϕ (xn ) = 6. We have four colors for xn . (2) cϕ (xn ) = 5. Let N (xn ) = {x1n , x2n , x3n , x4n , x5n , x6n }. W.l.o.g. suppose that ϕ(x1n ) = ϕ(x2n ) = 1, ϕ(x3n ) = 2, ϕ(x4n ) = 3, ϕ(x5n ) = 4 and ϕ(vn6 ) = 5. If one of 6, 7, 8, 9 and 10 is not in Cϕ (x1n ) ∩ Cϕ (x2n ), we can color xn with this color to get an acyclic 10-coloring of G. Suppose {6, 7, 8, 9, 10} = Cϕ (x1n ) ∩ Cϕ (x2n ). If there exists k(2 ≤ k ≤ 5) such that Cϕ (xkn+1 ) ̸= {6, 7, 8, 9, 10}, we recolor x1n with k(k ∈ {2, 3, 4, 5}) and color xn with a color in {6, 7, 8, 9, 10} \ Cϕ (xkn+1 ) to get a acyclically 10-coloring of G. If for all 2 ≤ k ≤ 5, Cϕ (xnk+1 ) = {6, 7, 8, 9, 10}, we recolor all the neighbors of xn with 1, color xn with 2 to get an acyclic 10-coloring of G. That completes the proof of Theorem 1.7. Remark 2. We believe that this bound is not tight and the best possible bound should be 7 as mentioned in Question 2 [11]. Acknowledgments The authors would like to express their sincere gratitude to the referees for their very careful reading of the paper and valuable suggestions that greatly improve this paper. The first author was supported by The Key Programs of Wuxi City College of Vocational Technology (WXCY-2012-GZ-007) and Research Foundation for Advanced Talents of Wuxi City College of Vocational Technology. The second author was supported by Fundamental Research Funds for the Central Universities (2012LWB46) and National Natural Science Foundation of China (11271365). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
M.O Albertson, D.M. Berman, Every planar graph has an acyclic 7-coloring, Israel J. Math. 28 (1977) 169–174. N. Alon, C. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Structures Algorithms 2 (3) (1991) 277–288. Béla Bollobás, Modern Graph Theory, Springer-Verlag, New York, Inc., 1998. O.V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979) 211–236. M.I. Burstein, Every 4-valent graph has an acyclic 5-coloring, Soob.S.C. Akad. Nauk Gruzin. SSR 93 (1979) 21–24 (in Russian). G. Fertin, E. Godard, A. Raspaud, Acyclic and k-distance coloring of the grid, Inform. Process. Lett. 87 (1) (2003) 51–58. G. Fertin, E. Godard, A. Raspaud, Minimum feedback vertex set and acyclic coloring, Inform. Process. Lett. 84 (2002) 131–139. G. Fertin, A. Raspaud, Acyclic coloring of graphs of maximum degree five: nine colors are enough, Inform. Process. Lett. 105 (2008) 65–72. A. Fiedorowicz, Acyclic 6-coloring of graphs with maximum degree 5 and small maximum average degree, Discuss. Math. Graph Theory 33 (2013) 91–99. B. Grünbaum, Acyclic colorings of planar graphs, Israel J. Math. 14 (3) (1973) 390–408. Hervé Hocquard, Graphs with maximum degree 6 are acyclically 11-colorable, Inform. Process. Lett. 111 (2011) 748–753. A.V. Kostochka, Acyclic 6-colorings of planar graphs, Metody Diskret. Anal. 28 (1976) 40–56 (in Russian). A.V. Kostochka, C. Stocker, Graphs with maximum degree 5 are acyclically 7-colorable, Ars Math. Contemp. 4 (2011) 153–164. J. Mitchem, Every planar graph has an acyclic 8-coloring, Duke Math. J. 41 (1974) 177–181. S. Skulrattanakulchai, Acyclic colorings of subcubic graphs, Inform. Process. Lett. 92 (2004) 161–167. ˘ S. Spacapan, A. Tepeh, On acyclic colorings of direct products, Discuss. Math. Graph Theory 28 (2008) 323–333. D. West, Introduction to Graph Theory, Prentice-Hall, 2001. K. Yadav, S. Varagani, K. Kothapalli, V.Ch. Venkaiah, Acyclic vertex coloring of graphs of maximum degree 6, Electron. Notes Discrete Math. 35 (2009) 177–182. K. Yadav, S. Varagani, K. Kothapalli, V.Ch. Venkaiah, Acyclic vertex coloring of graphs of maximum degree 5, Discrete Math. 311 (2011) 342–348.
Further reading [1] J. Beck, An alorithmic approach to the Lovasz Local Lemma, Random Structures Algorithms 2 (1991) 343–365.
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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Acyclic vertex coloring of graphs of maximum degree six Yancai Zhao a , Lianying Miao b,∗ , Shiyou Pang b , Wenyao Song b a
Wuxi City College of Vocational Technology, Wuxi, Jiangsu, 221116, PR China
b
School of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, PR China
article
info
Article history: Received 31 July 2013 Received in revised form 22 January 2014 Accepted 27 January 2014 Available online 4 March 2014
abstract In this paper, we prove that every graph with maximum degree six is acyclically 10colorable, thus improving the main result of Hervé Hocquard (2011). © 2014 Published by Elsevier B.V.
Keywords: Graph coloring Bounded degree graphs Acyclic coloring
1. Introduction A proper vertex coloring of a graph G = (V , E ) is an assignment of colors to the vertices of G such that two adjacent vertices do not use the same color. A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles; in other words, the graph induced by every two color classes is a forest. The acyclic chromatic number of G, denoted by χa (G), is the smallest integer k such that G is acyclically k-colorable. Acyclic colorings were introduced by Grünbaum [10]. The following are some results about acyclic colorings of graphs. Theorem 1.1 ([10]). Every planar graph is acyclically 9-colorable. Theorem 1.2 ([4]). Every planar graph is acyclically 5-colorable. This bound is tight since there exist 4-regular planar graphs [10] which are not acyclically 4-colorable. Theorem 1.3 ([2]). Every graph with maximum degree ∆ can be acyclically colored using O(∆(G)4/3 ) colors. Theorem 1.4 ([1]). Every graph with maximum degree ∆ can be acyclically colored using ∆(∆ − 1) + 2 colors. For graphs with maximum degree six, there are the following results. Theorem 1.5 ([18]). Every graph of maximum degree 6 can be acyclically colored with 12 colors. Theorem 1.6 ([11]). Every graph of maximum degree 6 can be acyclically colored with 11 colors. Other results about the acyclic coloring of graphs can be seen in [1,5,8,6,7,9–16,19]. Here we improve Theorem 1.6 by proving that. Theorem 1.7. Every graph with maximum degree six is acyclically 10-colorable.
∗
Corresponding author. E-mail addresses: [email protected] (Y. Zhao), [email protected] (L. Miao).
http://dx.doi.org/10.1016/j.disc.2014.01.022 0012-365X/© 2014 Published by Elsevier B.V.
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Y. Zhao et al. / Discrete Mathematics 325 (2014) 17–22
Fig. 1. An illustration of NC (u), nC (u), Cϕ (u) and cϕ (u).
This theorem also answers the second question posed by Hervé Hocquard [11]. We now introduce the notations (some of them are first given in [11]) and use the standard graph theory terminology [17] not defined here. Let G = (V (G), E (G)), and v ∈ V (G). We use N (v) and d(v) to denote the set of the neighbors and the degree of v in G respectively. A partial acyclic coloring ϕ of G is an assignment of colors to a subset U of V (G) such that ϕ is an acyclic coloring of G[U ]. Let ϕ be a partial acyclic coloring of G with the color set C and the colored subset U ⊆ V (G) and let v be an uncolored vertex of G. We say that a color c ∈ C is available for v if no neighbor of v is colored c. We say that a color c ∈ C is feasible for v if it is available for v and coloring v with c results in a partial acyclic coloring of G. We say that a color c ∈ C is no-feasible for v if it is available for v and coloring v with c results in bicolored cycles in G. Let Fv and NFv denote the set of feasible and no-feasible colors for v . For a vertex u ∈ V (G) (colored or uncolored), we denote the set and the number of colored neighbors of u by NC (u) = N (u) ∩ U and nC (u) = |NC (u)| respectively. We denote by Cϕ (u) the set of colors used by vertices in NC (u) and cϕ (u) = |Cϕ (u)|. For example, in Fig. 1, NC (u) = {v1 , v2 , v3 , v4 , v5 }, nC (u) = 5, Cϕ (u) = {1, 2, 3, 4} and cϕ (u) = 4. Finally, we denote by ∆(G), the maximum degree of a graph G. We assume that the graphs in this paper are connected. Let C = {1, 2, . . . , 10}. 2. Main result It is known that [3, P34] every graph of maximum degree at most ∆ is an induced subgraph of a ∆-regular graph, and it is sufficient to consider 6-regular connected graphs in this paper. The following definition is first given in [11]. Let G be a ∆-regular connected graph. A good spanning tree of G is a spanning tree T such that T contains a vertex adjacent to ∆ − 1 leaves. Lemma 1 ([11]). Every regular connected graph admits a good spanning tree. Remark 1. The idea of the proof of Theorem 1.7 is mainly from [11]. We make more careful analysis and use one new technique of constructing bipartite graphs to reduce the number of colors needed to 10. Proof of Theorem 1.7. Let G be a 6-regular connected graph. Let T be a good spanning tree of G. Let xn be a vertex adjacent to five leaves x1 , x2 , x3 , x4 , x5 in T . We order the vertices of G from x1 to xn according to a post-order walk of T . First, we color x1 , x2 , x3 , x4 , x5 with five distinct colors. Then we will successively color x6 , x7 , . . . , xn−1 while the colors of x1 , x2 , x3 , x4 , x5 will never be changed. Finally, we color xn . Suppose that we have colored x1 , x2 , . . . , xi−1 (6 ≤ i ≤ n − 1). Let ϕ be an acyclic 10-coloring of Gi−1 = G[x1 , x2 , . . . , xi−1 ]. Now we color xi = u. Since u is adjacent to at least one of xi+1 , . . . , xn , we have nC (u) ≤ 5. W.l.o.g. j assume that nC (u) = 5. Let NC (u) = {v1 , v2 , v3 , v4 , v5 }. For 1 ≤ i, j, k ≤ 5, let N (vi )\{u} = {vi |1 ≤ j ≤ 5} and j
j ,k
N (vi )\{vi } = {vi |1 ≤ k ≤ 5}. Let A = {x1 , x2 , x3 , x4 , x5 }. (See Fig. 2.) Since u ̸= xn , we have the following claim.
Claim 1. If v ∈ N (u) and nC (v) = 5, then v ̸∈ A. If v ̸∈ N (u) and nC (v) = 6, then v ̸∈ A. Construction. We construct a bipartite graph H with the bipartition (X , Y ) such that X = {x|x ∈ NC (u) and there is a vertex x′ ∈ NC (u) such that ϕ(x) = ϕ(x′ ) and x ̸= x′ } and Y = NFu . For any x ∈ X and y ∈ Y , x is adjacent to y in H iff assigning u the color y will result in a bicolored cycle passing through u and x. It is easy to see that dH (y) ≥ 2 for any y ∈ Y if X ̸= φ and Y ̸= φ. Now we consider the following five cases. Case 1. cϕ (u) = 5. Then there remain five colors for u. Case 2. cϕ (u) = 4. W.l.o.g. assume that ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = 2, ϕ(v4 ) = 3, ϕ(v5 ) = 4. We can color u with a color in C \{Cϕ (u) ∪ [Cϕ (v1 ) ∩ Cϕ (v2 )]} (which is not φ ).
Y. Zhao et al. / Discrete Mathematics 325 (2014) 17–22
j
19
j,k
Fig. 2. An illustration of vi , vi , vi .
Case 3. cϕ (u) = 3. Then exactly one color or two colors are repeated among the neighbors of u. If it is the former, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = 2, ϕ(v5 ) = 3. Otherwise, we can assume that ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = ϕ(v4 ) = 2, ϕ(v5 ) = 3. Case 3.1. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = 2, ϕ(v5 ) = 3. If Fu ̸= φ , we are done. Otherwise NFu = C \{ 1, 2, 3}. In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 } and Y = NFu . Since y∈Y dH (y) ≥ 14, at least one of v1 , v2 , v3 , say v1 , has dH (v1 ) = 5, that is cϕ (v1 ) = 5. By Claim 1 we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. Case 3.2. ϕ(v1 ) = ϕ(v2 ) = 1, ϕ(v3 ) = ϕ(v4 ) = 2, ϕ(v5 ) = 3. If Fu ̸= φ , we are done. Assume that Fu = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 } and Y = C \{1, 2, 3}. If one of v1 , v2 , v3 , v4 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 4, and then dH (vi ) ≤ 4 for all 1 ≤ i ≤ 4. Since y∈Y dH (y) ≥ 14 and dH (v1 ) = dH (v2 ), dH (v3 ) = dH (v4 ), so either dH (v1 ) = dH (v2 ) = 4 or dH (v3 ) = dH (v4 ) = 4, say dH (v1 ) = dH (v2 ) = 4. We can assume that v1 ̸∈ A. If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2, 3 and those in Cϕ (v1 ), and then it becomes Case 2. Assume nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 4, ϕ(v13 ) = 5, ϕ(v14 ) = 6, ϕ(v15 ) = 7. Then {8, 9, 10} ⊆ Cϕ (v3 ) ∩ Cϕ (v4 ) since Fu = φ . If one of 2, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), we can recolor v1 with this color, then it becomes Case 2 or Case 3.1. Assume {2, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since dH (v1 ) = 4, 1 is repeated among the neighbors of v11 or v12 , by Claim 1 we can recolor v11 or v12 with 3 and recolor v1 with 8, and then it becomes Case 2. Case 4. cϕ (u) = 2. Then exactly one color or two colors are repeated among the neighbors of u. If it is the former, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ) = 1, ϕ(v5 ) = 2. Otherwise, we can assume that ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = ϕ(v5 ) = 2. Case 4.1. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ) = 1, ϕ(v5 ) = 2. If one of v1 , v2 , v3 , v4 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), and then it becomes Case 3. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 4. If F ̸= φ , we are done. Assume that F = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 } and Y = C \{1, 2}. Clearly y∈Y dH (y) ≥ 16 and dH (vi ) ≤ 4 for all 1 ≤ i ≤ 4. Thus dH (x) = 4 for any x ∈ X and dH (y) = 2 for any y ∈ Y . Since ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = ϕ(v4 ), at least one of vi′ s(1 ≤ i ≤ 4) is not in A, say v1 . If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), then it becomes Case 3. Assume that nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 3, ϕ(v13 ) = 4, ϕ(v14 ) = 5 and ϕ(v15 ) = 6. If one of 7, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), say 7, we can recolor v1 with 7, then it becomes Case 3. Suppose {7, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since 3 ̸∈ Fu and dH (v1 ) = 4, there will be a bicolored cycle colored 1 and 3 passing through u, v1 and v11 or v12 if we color u with 3, so at least one of v11 and v12 has a neighbor (other than v1 ) colored 1, say v11 . By Claim 1 we can recolor v11 with 2 and recolor v1 with 7, and then it becomes Case 3. Case 4.2. ϕ(v1 ) = ϕ(v2 ) = ϕ(v3 ) = 1, ϕ(v4 ) = ϕ(v5 ) = 2. If one of v1 , v2 , v3 , v4 , v5 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), and then it becomes Case 3. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 5. If Fu ̸= φ , we are done. Assume that Fu = φ . In this case, graph H in the Construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 , v5 } and Y = C \{1, 2}. Note that dH (vi ) ≤ 4 for all 1 ≤ i ≤ 5.
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Fig. 3. An illustration of above assumption, ◦ uncolored, ⊗ colored or uncolored.
Suppose one of v1 , v2 , v3 , v4 , v5 , say v1 , has dH (v1 ) = 4 and v1 ̸∈ A. If nC (v1 ) = 4, we can recolor v1 with a color different from 1, 2 and those in Cϕ (v1 ), then it becomes Case 3. Assume that nC (v1 ) = 5 and w.l.o.g. assume that ϕ(v11 ) = ϕ(v12 ) = 3, ϕ(v13 ) = 4, ϕ(v14 ) = 5 and ϕ(v15 ) = 6. If one of 7, 8, 9, 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ), we can recolor v1 with this color, then it becomes Case 3. Suppose {7, 8, 9, 10} ⊆ Cϕ (v11 ) ∩ Cϕ (v12 ). Since 3 ̸∈ Fu and dH (v1 ) = 4, there will be a bicolored cycle colored 1 and 3 passing through u, v1 and v11 or v12 if we color u with 3, so there is one neighbor (other than v1 ) of v11 or v12 that colored 1, say v11 . Then by Claim 1 we can recolor v11 with 2 and recolor v1 with 7, and then it becomes Case 3. So assume that for all 1 ≤ i ≤ 5 either dH (vi ) ≤ 3 or dH (vi ) = 4 and vi ∈ A. Note that y∈Y dH (y) ≥ 16, dH (v4 ) = dH (v5 ) and at least one of v4 and v5 is not in A. We now assume that dH (v1 ) = 4, v1 ∈ A and dH (v2 ) = dH (v3 ) = 3, dH (v4 ) = dH (v5 ) = 3. Then y∈Y dH (y) = 16 and dH (y) = 2 for any y ∈ Y . We can assume that NH (v1 ) = {3, 4, 5, 6}, NH (v2 ) = {3, 4, 7}, NH (v3 ) = {5, 6, 7}, NH (v4 ) = NH (v5 ) = {8, 9, 10} and v4 ̸∈ A. Now we first consider v4 . If nC (v4 ) = 3, we can recolor v4 with 7, then it becomes Case 3. If nC (v4 ) = 4, we consider two cases: if cϕ (v4 ) = 4, we can recolor v4 with a color different from 1, 2 and those in Cϕ (v4 ), then it becomes Case 3. If cϕ (v4 ) = 3, then we may assume that ϕ(v41 ) = ϕ(v42 ) = 8. Then at least one of 1, 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ). We can recolor v4 with this color, and then it becomes Case 3 or Case 4.1. If nC (v4 ) = 5, we consider two cases: Case 4.2.I. cϕ (v4 ) = 4. W.l.o.g. assume that ϕ(v41 ) = ϕ(v42 ) = a. First suppose a ∈ {1, 3, 4, 5, 6, 7}, say a = 1 (the other cases are similar). If one of 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ). We can recolor v4 with this color, then it becomes Case 3. If {3, 4, 5, 6, 7} ⊆ Cϕ (v41 ) ∩ Cϕ (v42 ), by Claim 1 we can recolor v41 with 2 and recolor v4 with 3, then it becomes Case 3. Now suppose a ∈ {8, 9, 10}, say a = 8. W.l.o.g. assume that Cϕ (v4 ) = {1, 8, 9, 10}. Since 8 ̸∈ Fu and dH (v4 ) = 3, there will be a bicolored cycle colored 2 and 8 passing through u, v4 and v41 or v42 if we color u with 8, so there is one neighbor (other than v4 ) of v41 or v42 that colored 2, then at least one of 3, 4, 5, 6, 7 is not in Cϕ (v41 ) ∩ Cϕ (v42 ), say 3. We recolor v4 with 3, then it becomes Case 3. Case 4.2.II. cϕ (v4 ) = 3, and then Cϕ (v4 ) = {8, 9, 10}. We have two cases to consider: exactly one color or two colors are repeated among the neighbors of v4 . Case 4.2.II-1 It is the former case. We can assume that ϕ(v41 ) = ϕ(v42 ) = ϕ(v43 ) = 8, ϕ(v44 ) = 9, ϕ(v45 ) = 10. Since dH (v4 ) = 3, there is one neighbor (other than v1 ) of v41 or v42 or v43 that colored 2, say v41 . Let Y1 = {1, 3, 4, 5, 6, 7}. We assume that recoloring v4 with any color in Y1 will result in a bicolored cycle, for otherwise it will become C4.1 or C3 after recoloring v4 with the color. Now similarly to H, we construct a bipartite graph H1 with the bipartition (X1 , Y1 ) such that X1 = {v41 , v42 , v43 }. For any x ∈ X1 and y ∈ Y1 , x is adjacent to y in H1 iff reassigning v4 the color y will result in a bicolored cycle passing through v4 and x. We know that dH1 (y) ≥ 2 for any y ∈ Y1 . If dH1 (x) = 5 for some x ∈ X1 , we recolor x with the color in {1, 3, 4, 5, 6, 7} \ Cϕ (x), then it becomes Case 4.2.I. So assume dH1 (x) ≤ 4 for any x ∈ X1 . By the size of H1 , dH1 (x) = 4 for any x ∈ X1 . W.l.o.g. assume that NH1 (v41 ) = {1, 3, 4, 5}. Note that v41 has two neighbors colored 2, so v41 ̸∈ A. By Claim 1 we can recolor v41 with 6 to get Case 4.2.I. Case 4.2.II-2 It is the latter case. W.l.o.g. assume that ϕ(v41 ) = ϕ(v42 ) = 8, ϕ(v43 ) = ϕ(v44 ) = 9, ϕ(v45 ) = 10. Note that dH (v4 ) = 3, one of v41 and v42 passes through the bicolored cycle colored 2 and 8 if we color u with 8, say v41 , that is v41 has a neighbor (other than v1 ) colored 2. Similarly, one of v43 and v44 passes through the bicolored cycle colored 2 and 9 if we color u with 9, say v43 , that is v43 has a neighbor (other than v1 ) colored 2. We assume that recoloring v4 with any color in Y1 will result in a bicolored cycle, for otherwise it will become C4.1 or C3 after recoloring v4 with the color. Now we construct a bipartite graph H2 with the bipartition (X2 , Y1 ) such that X2 = {v41 , v42 , v43 , v44 }. For any x ∈ X2 and y ∈ Y1 , x is adjacent to y in H2 iff reassigning v4 color y will result in a bicolored cycle passing through v4 and x. (See Fig. 3.) We know that dH2 (y) ≥ 2 for any y ∈ Y1 . If dH2 (x) = 5 for some x ∈ X2 , we recolor x with the color in {1, 3, 4, 5, 6, 7} \ Cϕ (x), then it becomes Case 4.2.I. So assume dH2 (x) ≤ 4 for any x ∈ X2 . If dH2 (v41 ) = 4 or dH2 (v43 ) = 4, say v41 . By Claim we can recolor v41 with a color not in Cϕ (v4 ) ∪ Cϕ (v41 ) to get Case 4.2.I. So assume dH2 (x) = 3 for any x ∈ X2 .
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21
Fig. 4. An illustration of above assumption, ◦ uncolored, ⊗ colored or uncolored.
Then dH2 (y) = 2 for any y ∈ Y2 . We assume that NH2 (v41 ) = NH2 (v42 ) = {1, 3, 4} and NH2 (v43 ) = NH2 (v44 ) = {5, 6, 7}. (See Fig. 4). If both v41 and v43 are in A, we consider v2 . If nC (v2 ) = 3, we can recolor v2 with 5, then it becomes Case 3. If nC (v2 ) = 4, we consider two cases: if cϕ (v2 ) = 4, we can recolor v2 with a color different from 1, 2 and those in Cϕ (v2 ), then it becomes Case 3. If cϕ (v2 ) = 3, then we may assume that ϕ(v21 ) = ϕ(v22 ) = 3. If one of 5, 6, 8, 9 and 10, say 5, is not in Cϕ (v21 ) ∩ Cϕ (v22 ). We can recolor v2 with 5, and then it becomes Case 3. If {5, 6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), by Claim 1 we can recolor v21 , v22 with 1 and recolor v2 with 3, then it becomes Case 3. If nC (v2 ) = 5, we consider two cases. Case 4.2.I′ . cϕ (v2 ) = 4. W.l.o.g. assume that ϕ(v21 ) = ϕ(v22 ) = a. We first consider the case that a = 2. If {5, 6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), we recolor v21 with 1, recolor v2 with 5, then it becomes Case 3. If {5, 6, 8, 9, 10} ̸⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), say 5 ̸∈ Cϕ (v21 ) ∩ Cϕ (v22 ), we recolor v2 with 5, then it becomes Case 3. If a ∈ {3, 4, 7}, w.l.o.g. we assume that a = 3. Since dH (v2 ) = 3 and NH (v2 ) = {3, 4, 7}, there will be a bicolored cycle colored 1 and 3 passing through u, v2 and v21 or v22 if we color u with 3, so there is one neighbor (other than v2 ) of v21 or v22 that colored 1, say v21 . If C \ {2} ̸= Cϕ (v2 ) ∪ Cϕ (v21 ), choose a color in [C \ {2}] \ [Cϕ (v2 ) ∪ Cϕ (v21 )] to color v2 , then it becomes Case 3. If C \ {2} = Cϕ (v2 ) ∪ Cϕ (v21 ), we can recolor v21 with 2 and recolor v2 with a color different from 1, 2 and those in ϕ(v2 ), then it becomes Case 3. If a ∈ {5, 6, 8, 9, 10}, say a = 5, then {6, 8, 9, 10} ⊆ Cϕ (v21 ) ∩ Cϕ (v22 ), for otherwise, we can recolor v2 with the color in {6, 8, 9, 10} \ [Cϕ (v21 ) ∩ Cϕ (v22 )], i,1
i,2
i,3
i,4
then it becomes Case 3. Let ϕ(v2 ) = 6, ϕ(v2 ) = 8, ϕ(v2 ) = 9, ϕ(v2 ) = 10 for i = 1, 2. If one of nC (v21 ) and nC (v22 ) is 6,
)∈ ̸ {6, 8, 9, 10}, we recolor v with a color α that is not in {6, 8, 9, 10, ϕ(v21,5 ), 3, 4, 5, 7} 1,5 1,5 and recolor v2 with a color not in {1, 2, 3, 4, 5, 7, α}, then it becomes Case 3. If ϕ(v2 ) ∈ {6, 8, 9, 10}, say v2 = 6. Now 1,1 1,5 1 ,1 1 ,5 1 1 we consider v2 , v2 and v2 . If 1 or 2 is not in ϕ(v2 ) ∩ ϕ(v2 ), we recolor v2 with 1 or 2 and recolor v2 with 6, then it 1,1 1 ,5 1 ,1 1,5 becomes Case 3. So we assume that {1, 2} ⊆ ϕ(v2 ) ∩ ϕ(v2 ). If one of 3, 4 and 7 is not in ϕ(v2 ) ∩ ϕ(v2 ), we recolor v21 1,1 1 ,5 1 ,1 with 3 this color, then it becomes the above case. So we assume that {3, 4, 7} ⊆ ϕ(v2 ) ∩ ϕ(v2 ). Then we recolor v2 and v21,5 with 5, and recolor v21 with 6, and recolor v2 with 8, and then it becomes Case 3. If nC (v21 ) = nC (v22 ) = 5. We assume that v21 ̸∈ A. We recolor v21 with 1 and recolor v2 with 6, and then it becomes Case 3. we assume nC (v ) = 6. If ϕ(v 1 2
1 ,5 2
1 2
Case 4.2.II′ . cϕ (v2 ) = 3. Similar to the above discussion in Case 4.2.II, either there are two neighbors of v2 , say v21 and v23 , which are in A or at least one of v21 and v23 is not in A. If both v21 and v23 are in A. We consider v3 . At the step that exactly two colors are repeated among the neighbors of v3 , all neighbors of v3 are not in A. So we can assume that one of v41 and v43 is not in A, say v41 . From now on we consider v41 . If the last neighbor of v41 is uncolored or colored a ̸∈ {1, 2, 3, 4}, we recolor v41 with one color different from those in Cϕ (v41 ) and those in Cϕ (v4 ). We obtain Case 4.2.I. Assume that the last neighbor of v41 is colored a ∈ {1, 2, 3, 4}. If the another neighbor of v43 is colored b ̸∈ {2, 5, 6, 7}, by Claim 1 we recolor v43 with one color different from those in Cϕ (v43 ) and those in Cϕ (v4 ), we obtain Case 4.2.I. Assume that the other neighbor of v43 is colored b ∈ {2, 5, 6, 7} or uncolored. 1 ,1
1,2
1 ,3
1,4
1 ,5
If a = 2, w.l.o.g. suppose ϕ(v4 ) = ϕ(v4 ) = 2, ϕ(v4 ) = 1, ϕ(v4 ) = 3 and ϕ(v4 ) = 4. Since dH (v4 ) = 3 and v41 1 ,1
passes through a bicolored cycle colored 2 and 8 after coloring u with 8, at least one of v4 than v
1 4)
1,1 4 .
colored 8, say v
1,2
and v4
has a neighbor (other
22
Y. Zhao et al. / Discrete Mathematics 325 (2014) 17–22 1,1
1 ,2
If one of 5, 6, 7 and 9 is not in Cϕ (v4 ) ∩ Cϕ (v4 ), we recolor v41 with this color. We obtain Case 4.2.I. If {5, 6, 7, 9} ⊆ 1 ,1 4
1,2 4
1,1 4
1,1
Cϕ (v ) ∩ Cϕ (v ), then Cϕ (v ) = {5, 6, 7, 8, 9}. By Claim 1 we recolor v4 with 10, recolor v41 with 5 and recolor v4 with 3. Then it becomes Case 3. 1,1 1,2 1,3 1,4 1,5 If a ∈ {1, 3, 4}, say a = 1. W.l.o.g. suppose ϕ(v4 ) = ϕ(v4 ) = 1, ϕ(v4 ) = 2, ϕ(v4 ) = 3 and ϕ(v4 ) = 4. 1 ,1
Since NH2 (v41 ) = {1, 3, 4}, at least one of v4 1 ,1 4
7 and 9 is not in Cϕ (v 1,1 4
{5, 6, 7, 9} ⊆ Cϕ (v
1,2 4
1,2
and v4
1,1
has a neighbor (other than v41 ) colored 8, say v4 . If one of 5, 6,
) ∩ Cϕ (v ), we recolor v41 with this color, then it becomes Case 4.2.I or Case 4.2.II-1. Assume ) ∩ Cϕ (v41,2 ). By Claim 1 we recolor v41,1 with 10, recolor v41 with 5 and recolor v4 with 3. Then it
becomes Case 3. Case 5. cϕ (u) = 1. W.l.o.g. we assume that ϕ(vi ) = 1 for 1 ≤ i ≤ 5. If one of v1 , v2 , v3 , v4 , v5 , say v1 , has cϕ (v1 ) = 5, then by Claim 1 we can recolor v1 with a color different from 1 and those in Cϕ (v1 ), and then it becomes Case 4.1. We assume that cϕ (vi ) ≤ 4 for all 1 ≤ i ≤ 5. If F ̸= φ , we are done. Assume that F = φ . In this case, the graph H in the construction has a bipartition (X , Y ) such that X = {v1 , v2 , v3 , v4 , v5 } and Y = C \{1}. Note that y∈Y dH (y) ≥ 18. We know that at least three of vi′ s(1 ≤ i ≤ 5) have dH (vi ) = 4, say v1 , v2 , v3 . W.l.o.g. we assume that v1 ̸∈ A. If nC (v1 ) = 4, then we can recolor v1 with a color different from 1 and those in Cϕ (v1 ), and then it becomes Case 4.1. So assume that nC (v1 ) = 5. W.l.o.g. we assume that ϕ(v11 ) = ϕ(v12 ) = 2, ϕ(v13 ) = 3, ϕ(v14 ) = 4, ϕ(v15 ) = 5. Since dH (v1 ) = 4, so at least one of v11 and v12 has a neighbor (other than v1 ) colored 1, so at least one of 6, 7, 8, 9 and 10 is not in Cϕ (v11 ) ∩ Cϕ (v12 ). We recolor v1 with this color, and then it becomes Case 4.1. Now we color xn . We need to consider two cases. (1) cϕ (xn ) = 6. We have four colors for xn . (2) cϕ (xn ) = 5. Let N (xn ) = {x1n , x2n , x3n , x4n , x5n , x6n }. W.l.o.g. suppose that ϕ(x1n ) = ϕ(x2n ) = 1, ϕ(x3n ) = 2, ϕ(x4n ) = 3, ϕ(x5n ) = 4 and ϕ(vn6 ) = 5. If one of 6, 7, 8, 9 and 10 is not in Cϕ (x1n ) ∩ Cϕ (x2n ), we can color xn with this color to get an acyclic 10-coloring of G. Suppose {6, 7, 8, 9, 10} = Cϕ (x1n ) ∩ Cϕ (x2n ). If there exists k(2 ≤ k ≤ 5) such that Cϕ (xkn+1 ) ̸= {6, 7, 8, 9, 10}, we recolor x1n with k(k ∈ {2, 3, 4, 5}) and color xn with a color in {6, 7, 8, 9, 10} \ Cϕ (xkn+1 ) to get a acyclically 10-coloring of G. If for all 2 ≤ k ≤ 5, Cϕ (xnk+1 ) = {6, 7, 8, 9, 10}, we recolor all the neighbors of xn with 1, color xn with 2 to get an acyclic 10-coloring of G. That completes the proof of Theorem 1.7. Remark 2. We believe that this bound is not tight and the best possible bound should be 7 as mentioned in Question 2 [11]. Acknowledgments The authors would like to express their sincere gratitude to the referees for their very careful reading of the paper and valuable suggestions that greatly improve this paper. The first author was supported by The Key Programs of Wuxi City College of Vocational Technology (WXCY-2012-GZ-007) and Research Foundation for Advanced Talents of Wuxi City College of Vocational Technology. The second author was supported by Fundamental Research Funds for the Central Universities (2012LWB46) and National Natural Science Foundation of China (11271365). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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Further reading [1] J. Beck, An alorithmic approach to the Lovasz Local Lemma, Random Structures Algorithms 2 (1991) 343–365.