具有限制双色子图的图的着色：I.非循环、星形和树宽着色
Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings
摘要：
我们证明了对任意固定整数$m \geq 1$，最大度$Δ$的图有一个用$O(Δ^{(m+1)/m})$颜色着色的图，其中每个连通的双色子图至多包含$m$边.这一结果统一了先前已知的上界的颜色的数量足够的某些类型的图着色，包括星形着色，其中$O(Δ^{3/2})$颜色足够，和非循环着色，其中$O(Δ^{4/3})$颜色足够。我们的证明使用了Alon，McDiarmid和Reed的概率方法。这个结果也给出了以前未知的上界，包括这样一个事实，即最大程度$Δ$的图有一个适当的着色与$O(Δ^{9/8})$颜色，其中每个双色子图是平面的，以及一个适当的着色与$O(Δ^{13/12})$颜色，其中每个双色子图有树宽最多$3$。
英文摘要：
We show that for any fixed integer $m \geq 1$, a graph of maximum degree $Δ$ has a coloring with $O(Δ^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(Δ^{3/2})$ colors suffice, and acyclic colorings, for which $O(Δ^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree $Δ$ has a proper coloring with $O(Δ^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(Δ^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
