论文标题
卡利图中的常规套装
Regular sets in Cayley graphs
论文作者
论文摘要
In a graph $Γ$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers.在文献中，$（0,1）$ - 完美的集合被称为完美代码，$（1,1）$ - 完美套件称为总完美代码。在本文中，我们证明，对于任何有限的$ g $，如果非平凡的普通亚组$ h $ $ g $的$ g $是$ g $的cayley图中的完美代码，那么它也是$（a，b）$ - 在某些$ g $的$ g $中的$ g $中的$ g $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ b $ | leqssl | $ 0 \ leqslant b \ leqslant | h | $，这样$ \ gcd（2，| h | -1）$ divides $ a $。本文也证明了涉及总完美代码的类似结果。
In a graph $Γ$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper.
