A special case of Vu’s conjecture: colouring nearly disjoint graphs of bounded maximum degree

IF 0.9 4区 数学 Q3 COMPUTER SCIENCE, THEORY & METHODS
Tom Kelly, Daniela Kühn, Deryk Osthus
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引用次数: 1

Abstract

Abstract A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$ , then the following holds. For every fixed $C$ , if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$ , then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$ . This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.
Vu猜想的一个特例:最大有界度的近不相交图的上色
如果一组图的每一对至少有一个顶点相交,则该组图是几乎不相交的。我们证明了如果$G_1, \dots, G_m$是最大度为$D$的近不相交图,则下列成立。对于每个固定的$C$,如果每个顶点$v \in \bigcup _{i=1}^m V(G_i)$最多包含在图表$G_1, \dots, G_m$的$C$中,则$\bigcup _{i=1}^m G_i$的(列表)色数最多为$D + o(D)$。这一结果证实了Vu猜想的一个特例,推广了有界最大度线性一致超图的表色指标上的Kahn界。事实上,这个结果适用于对应色数(或DP),因此暗示了Molloy和Postle最近的一个结果,我们从“色度”设置中更一般的列表着色结果推导出这个结果,这个结果也暗示了Reed和Sudakov的结果。
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来源期刊
Combinatorics, Probability & Computing
Combinatorics, Probability & Computing 数学-计算机:理论方法
CiteScore
2.40
自引率
11.10%
发文量
33
审稿时长
6-12 weeks
期刊介绍: Published bimonthly, Combinatorics, Probability & Computing is devoted to the three areas of combinatorics, probability theory and theoretical computer science. Topics covered include classical and algebraic graph theory, extremal set theory, matroid theory, probabilistic methods and random combinatorial structures; combinatorial probability and limit theorems for random combinatorial structures; the theory of algorithms (including complexity theory), randomised algorithms, probabilistic analysis of algorithms, computational learning theory and optimisation.
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