Total coalitions of cubic graphs of order at most 10

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Document Type : Original paper

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1 Novosibirsk State University, Pirogova str. 2, Novosibirsk, 630090, Russia

2 Sobolev Institute of Mathematics, Ak. Koptyug av. 4, Novosibirsk, 630090, Russia

Abstract

A total coalition in a graph $G=(V,E)$ consists of two disjoint sets of vertices $V_{1}$ and $V_{2}$, neither of which is a total dominating set but whose union $V_{1}\cup V_{2}$, is a total dominating set. A total coalition partition in a 
graph $G$ of order $n=|V|$ is a vertex partition $\tau = \{V_1, V_2, \dots , V_k \}$ such that every set $V_i \in \tau$ is not a total dominating set but forms a total coalition with another set $V_j\in \tau$ which is not a total dominating set. 
The total coalition number $TC(G)$ equals the maximum $k$ of a total coalition partition of $G$. In this paper, we determine the total coalition number of all cubic graphs of order $n \le 10$.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 30 January 2024

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