2-Coupon Coloring of Cubic Graphs Containing 3-Cycle or 4-Cycle

TL;DR

This paper investigates the partitioning of cubic graphs with small cycles into two total dominating sets, providing counterexamples for 3-cycles and confirming the property for 4-cycles, extending previous results.

Contribution

It disproves a conjecture for cubic graphs with 3-cycles and proves the property holds for 4-cycles, advancing understanding of total domination in cubic graphs.

Findings

Counterexamples exist for cubic graphs with 3-cycles.

Cubic graphs with 4-cycles can be partitioned into two total dominating sets.

Connected cubic graphs with a diamond subgraph can be similarly partitioned.

Abstract

Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$. Recently, the following question was proposed: "Is it true that every connected cubic graph containing a $3$-cycle has two vertex disjoint total dominating sets?" In this paper, we give a negative answer to this question. Moreover, we prove that if we replace $3$-cycle with $4$-cycle the answer is affirmative. This implies every connected cubic graph containing a diamond (the complete graph of order $4$ minus one edge) as a subgraph can be partitioned into two total dominating sets, a result that was proved in 2017.