On cubic graphs having the maximal coalition number

Abstract

A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is called a coalition partition $\mathcal{P}$ if every non-dominating set of $\mathcal{P}$ is a member of a coalition and every dominating set is a single-vertex set. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partition. It is known that for cubic graphs $C(G)\le 9$. The existence of cubic graphs with the maximal coalition number is an unsolved problem. In this paper, an infinite family of cubic graphs satisfying $C(G)=9$ is constructed.