AI 中文总结
本文研究图的模积的独立数，通过刻画其独立集结构得到精确结果，给出若干紧界与精确结果，并引入划分图为构造独立集提供框架。
AI 中文摘要
图G和H的模积（modular product）G◇H是顶点集为V(G)×V(H)的图，其中两个顶点(g,h)与(g',h')相邻当且仅当满足以下任一条件：g=g'且hh'∈E(H)；gg'∈E(G)且h=h'；gg'∈E(G)且hh'∈E(H)；或当g≠g'且h≠h'时，gg'∉E(G)且hh'∉E(H)。图G的独立数α(G)是G中两两不相邻顶点构成集合的最大基数。本文研究图的模积的独立数，首先对G◇H的所有独立集进行结构刻画，从而得到α(G◇H)的精确结果；该结果的特殊情形给出了α(G◇H)的若干紧界及部分精确结果；最后，引入与G◇H关联的划分图，为从其子结构的独立集构造模积的独立集提供了框架。
英文摘要
The \emph{modular product} $G\diamond H$ of graphs $G$ and $H$ is a graph on vertex set $V(G)\times V(H)$. Two vertices $(g,h)$ and $(g',h')$ of $G\diamond H$ are adjacent if $g=g'$ and $hh'\in E(H)$, or $gg'\in E(G)$ and $h=h'$, or $gg'\in E(G)$ and $hh'\in E(H)$, or (for $g\neq g'$ and $h\neq h'$) $gg'\notin E(G)$ and $hh'\notin E(H)$. The independence number $α(G)$ of a graph $G$ is the maximum cardinality of a set of pairwise nonadjacent vertices in $G$. In this paper, we study the independence number of the modular product of graphs. We first structurally characterize all independent set of $G\diamond H$ which lead to the exact result on $α(G \diamond H)$. Special cases of this result lead to several sharp bounds and some exact results for $α(G \diamond H)$. Finally, we introduce a partition graph associated with $G \diamond H$ that provides a framework for constructing independent sets of the modular product from independent sets of its substructures.
Comments19 pages, two figures, 26 references
