An overview of (k,t)-regular sets and their applications

Abstract

A (k,t)-regular set is a vertex subset S inducing a k-regular subgraph such that every vertex out of S has t neighbors in S. This article is an expository overview of the main results obtained for graphs with (k,t)-regular sets. The graphs with classical combinatorial structures, like perfect matchings, Hamilton cycles, efficient dominating sets, etc, are characterized by (k,t)-regular sets whose determination is equivalent to the determination of those classical combinatorial structures. The characterization of graphs with these combinatorial structures are presented. The determination of (k,t)-regular sets in a finite number of steps is deduced and the main spectral properties of these sets are described.