Coalition of cubic graphs of order at most $10$

Abstract

The coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$, neither of which is a dominating set but whose union $V_1\cup V_2$, is a dominating set. A coalition partition in a graph $G$ is a vertex partition $\pi$ = $\{V_1,V_2,...,V_k\}$ such that every set $V_i\in \pi$ is not a dominating set but forms a coalition with another set $V_j\in \pi$ which is not a dominating set. The coalition number $C(G)$ equals the maximum $k$ of a coalition partition of $G$. In this paper, we compute the coalition number of all cubic graphs of order at most $10$.

arXiv comment14 pages, 3 figures