带颜色偏好的图着色

中文摘要 AI 辅助

我们研究带颜色偏好的图着色问题，其中每个顶点对可用颜色进行排序。除了为相邻顶点分配不同颜色外，我们还要求该着色是稳定的：不存在任何一组顶点能够循环交换它们被分配的颜色，使得每个顶点都严格偏好其新颜色甚于原颜色。我们将图G的稳定色数χ_stable(G)定义为满足任意偏好配置都存在稳定k-着色的最小整数k。我们确定了若干上下界，特别地，对于图G边的任意无环定向，从一个顶点出发通过有向路径可达的顶点最大数量（包含该顶点自身）是χ_stable(G)的一个上界，这表明χ_stable(G)是良定义的。我们还证明，对于树宽为t的n顶点图G，O(t log(1+n/t))种颜色足够，并以Grundy数给出相应下界。针对给定配置求最小稳定着色的问题，我们证明稳定2-着色性是多项式时间可解的，而对每个固定k≥3，稳定k-着色性是NP完全的。利用树宽界，我们给出以树宽为参数的固定参数可处理算法。

英文摘要

We study graph coloring with color preferences, in which each vertex ranks the available colors. In addition to assigning different colors to adjacent vertices, we require the coloring to be stable: no group of vertices can cyclically exchange their assigned colors so that each strictly prefers its new color to its original one. We define the stable chromatic number $χ_\mathrm{stable}(G)$ of a graph $G$ as the minimum integer $k$ such that every preference profile admits a stable $k$-coloring of $G$. We establish several upper and lower bounds. In particular, for any acyclic orientation of the edges of $G$, the largest number of vertices reachable from a vertex by directed paths, including the vertex itself, is an upper bound on $χ_\mathrm{stable}(G)$. This shows that $χ_\mathrm{stable}(G)$ is well-defined. We also show that $O(t \log (1+n/t))$ colors suffice for an $n$-vertex graph $G$ of treewidth $t$, and complement this with a lower bound in terms of the Grundy number. Turning to the problem of finding a minimum stable coloring for a given profile, we show that stable $2$-colorability is polynomial-time solvable, whereas stable $k$-colorability is NP-complete for every fixed $k\ge 3$. Using the treewidth bound, we give a fixed-parameter tractable algorithm parameterized by treewidth.

发表机构

• The University of Tokyo（东京大学）

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