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A NOTE ON REGULAR SETS IN CAYLEY GRAPHS
Published online by Cambridge University Press: 09 February 2023
Abstract
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.
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- Research Article
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
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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