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Published online by Cambridge University Press: 09 February 2023
A 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.
The first author was supported by the Natural Science Foundation of Chongqing (CSTB2022NSCQ- MSX1054) and the Foundation of Chongqing Normal University (21XLB006).
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