具有Grundy数$r+1$的$r$ -正则图族
On the family of $r$-regular graphs with Grundy number $r+1$
摘要：
图$G$的Grundy数，记为$Γ(G)$，是最大的$k$，使得对每个$j < i$，存在$V(G)$到$k$独立集$V_1,\ldots, V_k$的划分，且$V_i$的每个顶点至少与$V_j$中的一个顶点相邻.本文研究的对象是$r$ -正则图族，使得$Γ(G) = r + 1$ .利用独立模的概念，给出了$r=3$的这一族的一个刻画。此外，我们确定在这个家庭的图类，特别是类$r$ -正则图没有诱导$C_4$，$r \le 4$。此外，我们的命题包含了部分Grundy数的结果。
英文摘要：
The Grundy number of a graph $G$, denoted by $Γ(G)$, is the largest $k$ such that there exists a partition of $V(G)$, into $k$ independent sets $V_1,\ldots, V_k$ and every vertex of $V_i$ is adjacent to at least one vertex in $V_j$, for every $j < i$. The objects which are studied in this article are families of $r$-regular graphs such that $Γ(G) = r + 1$. Using the notion of independent module, a characterization of this family is given for $r=3$. Moreover, we determine classes of graphs in this family, in particular the class of $r$-regular graphs without induced $C_4$, for $r \le 4$. Furthermore, our propositions imply results on partial Grundy number.
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