Cubic Graphs with Total Domatic Number at Least Two

TL;DR

This paper characterizes cubic graphs that can be partitioned into at least two total dominating sets, advancing understanding of their structural properties and domination parameters.

Contribution

It provides a characterization of cubic graphs with a total domatic number of at least two, a problem previously not fully understood.

Findings

Identifies conditions for cubic graphs to have total domatic number ≥ 2

Provides structural insights into total dominating sets in cubic graphs

Advances theoretical understanding of domination in regular graphs

Abstract

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of $G$. In this paper we would like to characterize the cubic graphs with total domatic number at least two.