arXiv 1606.090622016-06-29math.CO

图的Anagram自由染色

Anagram-free colorings of graphs

Nina Kamčev, Tomasz Łuczak, Benny Sudakov

摘要：

如果序列$S$不包含连续符号$r_1 r_2\dots r_k r_{k+1} \dots r_{2k}$，使得$r_{k+1} \dots r_{2k}$是块$r_1 r_2\dots r_k$的置换，则该序列被称为无变位。为了解决埃尔德什和布朗的问题，Keränen在四个符号上构造了一个无限的无字谜序列。阿隆，Grytczuk，Hauszczak和Riordan的工作的动机，我们认为一个自然的广义的字谜免费序列图着色。给定图$G$的顶点着色称为无交错着色，如果$G$中任一条路上的颜色序列是无交错着色.我们称这种着色所需的最少颜色数为$G$的变位色数.本文研究了树、无次图和有界度图等几类图的变位色数。令人惊讶的是，我们发现，有界度图（如随机正则图），其中的字谜不能避免，除非我们基本上给每个顶点一个单独的颜色。

英文摘要：

A sequence $S$ is called anagram-free if it contains no consecutive symbols $r_1 r_2\dots r_k r_{k+1} \dots r_{2k}$ such that $r_{k+1} \dots r_{2k}$ is a permutation of the block $r_1 r_2\dots r_k$. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan, we consider a natural generalisation of anagram-free sequences for graph colorings. A coloring of the vertices of a given graph $G$ is called anagram-free if the sequence of colors on any path in $G$ is anagram-free. We call the minimal number of colors needed for such a coloring the anagram-chromatic number of $G$. In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we basically give each vertex a separate color.

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摘要：

英文摘要：

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