4-非正则平面图的2-距离着色
2-Distance Coloring of 4-Irregular Planar Graphs
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中文总结 AI 辅助
本文研究4-非正则平面图的2-距离着色问题，将其2-距离色数上界从Zhu证明的13改进为10，推进了平面图着色相关理论的研究。
中文摘要 AI 辅助
k-非正则图是指最大度为k且度为k的顶点互不相邻的图。图的2-距离k-着色是指用k种颜色对顶点着色，使得任意两个距离不超过2的顶点颜色不同。图G的2-距离色数记为χ₂(G)，是使得G存在2-距离k-着色的最小整数k。Zhu证明了最大度不超过4的平面图满足χ₂(G)≤13，本文证明对于4-非正则平面图G，χ₂(G)≤10。
英文摘要
A $k$-irregular graph is a graph with maximum degree $k$ such that vertices of degree $k$ are not adjacent. A $2$-distance $k$-coloring of a graph is a coloring of the vertices using $k$ colors in which any two vertices at distance at most $2$ receive distinct colors. The $2$-distance chromatic number of $G$, denoted by $χ_{2}(G)$, is the minimum integer $k$ such that $G$ admits a $2$-distance $k$-coloring. Zhu \cite{zhu} proved that $χ_2(G)\leq 13$ for planar graphs with maximum degree at most $4$. We prove that for a 4-irregular planar graph $G$, we have $χ_2(G) \leq 10$.
