A NOTE ON REGULAR SETS IN CAYLEY GRAPHS

Abstract

AbstractA subset R of the vertex set of a graph $\Gamma $ is said to be $(\kappa ,\tau )$ -regular if R induces a $\kappa $ -regular subgraph and every vertex outside R is adjacent to exactly $\tau $ vertices in R. In particular, if R is a $(\kappa ,\tau )$ -regular set of some Cayley graph on a finite group G, then R is called a $(\kappa ,\tau )$ -regular set of G. Let H be a nontrivial normal subgroup of G, and $\kappa $ and $\tau $ a pair of integers satisfying $0\leq \kappa \leq |H|-1$ , $1\leq \tau \leq |H|$ and $\gcd (2,|H|-1)\mid \kappa $ . It is proved that (i) if $\tau $ is even, then H is a $(\kappa ,\tau )$ -regular set of G; (ii) if $\tau $ is odd, then H is a $(\kappa ,\tau )$ -regular set of G if and only if it is a $(0,1)$ -regular set of G.