On regular sets in Cayley graphs
Xiaomeng Wang, Shou-Jun Xu, Sanming Zhou

TL;DR
This paper investigates the structure and existence of (a, b)-regular sets, especially (0, k)-regular sets, in Cayley graphs, including conditions for normal subgroups and specific groups like dihedral and quaternion groups.
Contribution
It characterizes when normal subgroups are (0, k)-regular sets and classifies subgroup (0, k)-regular sets in certain groups, extending known perfect code results.
Findings
Normal subgroups can be (0, k)-regular sets under specific conditions.
Complete classification of subgroup (0, k)-regular sets in dihedral and quaternion groups.
Conditions for (0, k)-regular sets in hypercubes and Cartesian products of cycles.
Abstract
Let be a graph and nonnegative integers. An -regular set in is a nonempty proper subset of such that every vertex in has exactly neighbours in and every vertex in has exactly neighbours in . A -regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset of a group is called an -regular set of if it is an -regular set in some Cayley graph of , and an -regular set in a Cayley graph of is called a subgroup -regular set if it is also a subgroup of . In this paper we study -regular sets in Cayley graphs with a focus on -regular sets, where is an integer. Among other things we determine when a non-trivial proper normal subgroup of a group is a -regular set of the…
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Taxonomy
Topicsgraph theory and CDMA systems · Finite Group Theory Research · Coding theory and cryptography
