论文标题
图形的无环着色
Acyclic colourings of graphs with obstructions
论文作者
论文摘要
给定图表$ g $，如果$ g $的颜色是$ g $的适当着色，并且每个周期都至少包含三种颜色。它的无环色数$χ_a（g）$是最低$ k $，因此存在适当的$ k $颜色$ g $，而没有任何双色周期。通常，当$ g $具有最高度$δ$时，众所周知，$χ_a（g）= o（δ^{4/3}）$作为$δ\ to \ infty $。我们研究了对这一进一步的效果，要求$ g $不包含$ t $顶点上的一些固定子图$ f $。我们确定，如果$ f $是细分树，$ o（t^{8/3}δ^{2/3}）$如果$ f $是森林，$ o（\ sqrt {t}Δ）$如果$ o（\ sqrt {t}Δ）如果$ f = k_ {3，t} $，$（t^{1/4}δ^{5/4}）$。
Given a graph $G$, a colouring of $G$ is acyclic if it is a proper colouring of $G$ and every cycle contains at least three colours. Its acyclic chromatic number $χ_a(G)$ is the minimum $k$ such that there exists a proper $k$-colouring of $G$ with no bicoloured cycle. In general, when $G$ has maximum degree $Δ$, it is known that $χ_a(G) = O(Δ^{4/3})$ as $Δ\to \infty$. We study the effect on this bound of further requiring that $G$ does not contain some fixed subgraph $F$ on $t$ vertices. We establish that the bound is constant if $F$ is a subdivided tree, $O(t^{8/3}Δ^{2/3})$ if $F$ is a forest, $O(\sqrt{t}Δ)$ if $F$ is bipartite and 1-acyclic, $2Δ+ o(Δ)$ if $F$ is an even cycle of length at least $6$, and $O(t^{1/4}Δ^{5/4})$ if $F=K_{3,t}$.
