环形网格的入射色数
The Incidence Chromatic Number of Toroidal Grids
摘要：
图$G$中的关联是具有$v \in V(G)$和$e \in E(G)$的$(v,e)$对，使得$v$和$e$是关联的。如果$v=w$或$e=f$或边缘$vw$等于$e$或$f$，则两个发生点$(v,e)$和$(w,f)$相邻。$G$的关联色数是存在从$G$的关联集到将不同颜色分配给相邻关联集的$k$颜色集的映射的最小$k$。本文证明了当$m,n \equiv 0 \pmod 5$为零时，环形网格$T_{m,n}=C_m\Box C_n$的关联色数为5，否则为6.
英文摘要：
An incidence in a graph $G$ is a pair $(v,e)$ with $v \in V(G)$ and $e \in E(G)$, such that $v$ and $e$ are incident. Two incidences $(v,e)$ and $(w,f)$ are adjacent if $v=w$, or $e=f$, or the edge $vw$ equals $e$ or $f$. The incidence chromatic number of $G$ is the smallest $k$ for which there exists a mapping from the set of incidences of $G$ to a set of $k$ colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid $T_{m,n}=C_m\Box C_n$ equals 5 when $m,n \equiv 0 \pmod 5$ and 6 otherwise.
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。
