Disjoint connected dominating sets in pseudorandom graphs

TL;DR

This paper proves that pseudorandom graphs with regular degree contain nearly the maximum possible number of disjoint connected dominating sets, highlighting their strong connectivity properties.

Contribution

It establishes the asymptotic number of disjoint CDSs in pseudorandom graphs, matching the known bounds for random regular graphs.

Findings

d-regular pseudorandom graphs contain approximately d/ln d disjoint CDSs

Random d-regular graphs typically have about d/ln d disjoint CDSs

The result is asymptotically optimal

Abstract

A connected dominating set (CDS) in a graph is a dominating set of vertices that induces a connected subgraph. Having many disjoint CDSs in a graph can be considered as a measure of its connectivity, and has various graph-theoretic and algorithmic implications. We show that $d$-regular (weakly) pseudoreandom graphs contain $(1+o(1))d/\ln d$ disjoint CDSs, which is asymptotically best possible. In particular, this implies that random $d$-regular graphs typically contain $(1+o(1))d/\ln d$ disjoint CDSs.