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A strong edge coloring of a graph is an assignment of colors to the edges of the graph such that for every color, the set of edges that are given that color form an induced matching in the graph. The strong chromatic index of a graph G, denoted by , is the minimum number of colors needed in any strong edge coloring of G. A graph is said to be chordless if there is no cycle in the graph that has a chord. Faudree, Gyárfás, Schelp, and Tuza (The Strong Chromatic Index of Graphs, Ars Combin 29B (1990), 205–211) considered a particular subclass of chordless graphs, namely, the class of graphs in which all the cycle lengths are multiples of four, and asked whether the strong chromatic index of these graphs can be bounded by a linear function of the maximum degree. Chang and Narayanan (Strong Chromatic Index of 2‐degenerate Graphs, J Graph Theory, 73(2) (2013), 119–126) answered this question in the affirmative by proving that if G is a chordless graph with maximum degree Δ, then . We improve this result by showing that for every chordless graph G with maximum degree Δ, . This bound is tight up to an additive constant. 相似文献
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We prove that the strong chromatic index of a 2‐degenerate graph is linear in the maximum degree Δ. This includes the class of all chordless graphs (graphs in which every cycle is induced) which in turn includes graphs where the cycle lengths are multiples of four, and settles a problem by Faudree et al. (Ars Combin 29(B) (1990), 205–211). © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 119–126, 2013 相似文献
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A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3.The star chromatic indexχ'st(G)of a graph G is the smallest integer k such that G has a star k-edge-coloring.The list star chromatic index ch'st(G)is defined analogously.The star edge coloring problem is known to be NP-complete,and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear.In this paper,we study,in contrast,the best linear upper bound for sparse graph classes.We show that for everyε>0 there exists a constant c(ε)such that if mad(G)<8/3-ε,then■and the coefficient 3/2 ofΔis the best possible.The proof applies a newly developed coloring extension method by assigning color sets with different sizes. 相似文献
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A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most 2 receive distinct colors. The strong chromatic index χ'_s(G) of a graph G is the minimum number of colors used in a strong edge coloring of G. In an ordering Q of the vertices of G, the back degree of a vertex x of G in Q is the number of vertices adjacent to x, each of which has smaller index than x in Q. Let G be a graph of maximum degree Δ and maximum average degree at most 2 k. Yang and Zhu [J. Graph Theory, 83, 334–339(2016)] presented an algorithm that produces an ordering of the edges of G in which each edge has back degree at most 4 kΔ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤ 4 kΔ-2 k + 1. In this note, we improve the algorithm of Yang and Zhu by introducing a new procedure dealing with local structures. Our algorithm generates an ordering of the edges of G in which each edge has back degree at most(4 k-1)Δ-2 k in the square of the line graph of G, implying that χ'_s(G) ≤(4 k-1)Δ-2 k + 1. 相似文献
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Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices of G. 相似文献
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A proper k-edge coloring of a graph G is an assignment of one of k colors to each edge of G such that there are no two edges with the same color incident to a common vertex. Let f(v) denote the sum of colors of the edges incident to v. A k-neighbor sum distinguishing edge coloring of G is a proper k-edge coloring of G such that for each edge uv∈E(G), f(u)≠f(v). By χ'_∑(G), we denote the smallest value k in such a coloring of G. Let mad(G) denote the maximum average degree of a graph G. In this paper, we prove that every normal graph with mad(G) ■ and Δ(G) ≥ 8 admits a(Δ(G) + 2)-neighbor sum distinguishing edge coloring. Our approach is based on the Combinatorial Nullstellensatz and discharging method. 相似文献
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对简单图G(V,E),f是从V(G)∪E(G)到{1,2,…,k}的映射,k是自然数,若f满足(1)uv,uw∈E(G),u≠w,f(uv)≠f(uw);(2)uv∈E(G),C(u)≠C(v).则称f是G的一个邻强边染色,最小的k称为邻强边色数,其中C(u)={f(uv)|uv∈E(G)}.给出了一类3-正则重圈图的邻强边色数. 相似文献
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研究了一类简单图G的色数x(G)与最大度△(G)的关系,对满足x(G)>(S~2+S)/2的X(G)+S阶色临界图G,证明了x(G)=△(G)+1-S,或等价地,△(G)+1-[((8△(G)+17~(1/2)-3/2]≤X(G)≤△(G)+1,这一结果部分改进了Brooks经典不等式X(G)≤△(G)+1,并完全刻画n+3(n≥4)个顶点的n-临界图的结构。 相似文献
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马刚 《数学的实践与认识》2012,42(9):207-213
研究了一些Mycielski图的点可区别均匀全染色(VDETC),利用构造法给出了路、圈、星和扇的Mycielski图的点可区别均匀全色数,验证了它们满足点可区别均匀全染色猜想(VDETCC). 相似文献
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For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than and maximum degree Δ at least 4 are 2‐distance ‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than (resp. , ) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance ‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than and with large enough maximum degree Δ (depending only on m) can be list 2‐distance ‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance ‐colored: the question of what happens between and 3 remains open. We prove also that any graph with maximum average degree can be list 2‐distance ‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than colors. Most of the above results can be transposed to injective list coloring with one color less. 相似文献
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The total chromatic number of series-parallel graphs of maximum degree greater than or equal to 4 will be determined using
the double inductions and the method of exchanging colors from the aspect of configuration property. Thus, the result of paper
[7] is a special case of this paper.
This research was supported by National Natural Science Foundation of China under grant NSFC60503002 相似文献
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关于图的星色数的一点注记 总被引:1,自引:0,他引:1
A star coloring of an undirected graph G is a proper coloring of G such that no path of length 3 in G is bicolored.The star chromatic number of an undirected graph G,denoted by χs(G),is the smallest integer k for which G admits a star coloring with k colors.In this paper,we show that if G is a graph with maximum degree △,then χs(G) ≤ [7△3/2],which gets better bound than those of Fertin,Raspaud and Reed. 相似文献
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References: 《高校应用数学学报(英文版)》2007,22(2):163-168
Let x(G^2) denote the chromatic number of the square of a maximal outerplanar graph G and Q denote a maximal outerplanar graph obtained by adding three chords y1 y3, y3y5, y5y1 to a 6-cycle y1y2…y6y1. In this paper, it is proved that △ + 1 ≤ x(G^2) ≤△ + 2, and x(G^2) = A + 2 if and only if G is Q, where A represents the maximum degree of G. 相似文献
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设G为2-连通平面图,若存在G的面f0,其中f0的边界构成的圈上无弦且V(f0)中的点的度至少为3,使得在G中去掉f0边界上的所有边后得到的图为除V(f0)中的点外度不小于3的树T,则称G为伪-Halin图;若V(f0)中的点全为3度点,则称G为Halin-图.本文研究了这类图的完备色数,并证明了对△(G)≥ 6的伪-Halin图 G有 XC(C)=△(G)+1.其中△(G)和XC(G)分别表示G的最大度和完备色数. 相似文献
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设G(V,E)是简单连通图,T(G)为图G的所有顶点和边构成的集合,并设C是k-色集(k是正整数),若T(G)到C的映射f满足:对任意uv∈E(G),有f(u)≠f(v),f(u)≠f(uv),f(v)≠f(uv),并且C(u)≠C(v),其中C(u)={f(u)}∪{f(uv)|uv∈E(G)}.那么称f为图G的邻点可区别E-全染色(简记为k-AVDETC),并称χ_(at)~e(G)=min{k|图G有k-邻点可区别E-全染色}为G的邻点可区别E-全色数.图G的中间图M(G)就是在G的每一个边上插入一个新的顶点,再把G上相邻边上的新的顶点相联得到的.探讨了路、圈、扇、星及轮的中间图的邻点可区别E-全染色,并给出了这些中间图的邻点可区别E-全色数. 相似文献