Anagram-free colourings of graph subdivisions
Anagram-free colourings of graph subdivisions
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图形细分的无字谜着色
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
D. Wood
中科院分区：
文献类型：
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作者：
Tim E. Wilson;D. Wood
变位词是$WP$形式的单词，其中$W$是非空单词，$P$是$W$的排列。如果图的子路径不是变位，则图的顶点着色是无变位的。无字谜图形着色是由Kamv{c}ev， {L}uczak和Sudakov和我们自己独立引入的。本文介绍了图细分的无字谜着色的研究。我们证明每个图都有一个无字谜的$8$可着色的细分。每条边的分割顶点数是边数的指数。对于树，我们构建无字谜的$10$可着色的细分，每条边的划分顶点更少。反过来，我们证明了完全图的细分和有界度的完全树的细分在无字型色数上的下界，即每条边的划分顶点。
An anagram is a word of the form $WP$ where $W$ is a non-empty word and $P$ is a permutation of $W$. A vertex colouring of a graph is anagram-free if no subpath of the graph is an anagram. Anagram-free graph colouring was independently introduced by Kamv{c}ev, {L}uczak and Sudakov and ourselves. In this paper we introduce the study of anagram-free colourings of graph subdivisions. We show that every graph has an anagram-free $8$-colourable subdivision. The number of division vertices per edge is exponential in the number of edges. For trees, we construct anagram-free $10$-colourable subdivisions with fewer division vertices per edge. Conversely, we prove lower bounds, in terms of division vertices per edge, on the anagram-free chromatic number for subdivisions of the complete graph and subdivisions of complete trees of bounded degree.