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

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 tota...

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Bibliographic Details
Main Authors Akbari, S, Azimian, M, Khani, A. Fazli, Samimi, B, Zahiri, E
Format Journal Article
LanguageEnglish
Published 29.08.2023
Subjects
Mathematics - Combinatorics
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Summary: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.
DOI:10.48550/arxiv.2308.15114