2-Coupon Coloring of Cubic Graphs Containing 3-Cycle or 4-Cycle
S. Akbari, M. Azimian, A. Fazli Khani, B. Samimi, and E. Zahiri

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 be a graph. A total dominating set in a graph is a set of vertices of such that every vertex in is adjacent to a vertex in . Recently, the following question was proposed: "Is it true that every connected cubic graph containing a -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 -cycle with -cycle the answer is affirmative. This implies every connected cubic graph containing a diamond (the complete graph of order minus one edge) as a subgraph can be partitioned into two total dominating sets, a result that was proved in 2017.
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Taxonomy
TopicsAdvanced Graph Theory Research
