The shortest cycle having the maximal number of coalition graphs

Abstract

A coalition in a graph G with a vertex set V consists of two disjoint sets V 1 , V 2 ⊂ V , such that neither V 1 nor V 2 is a dominating set, but the union V 1 ∪ V 2 is a dominating set in G . A partition of V is called a coalition partition π if every non-dominating set of π is a member of a coalition and every dominating set is a single-vertex set. Every coalition partition generates its coalition graph. The vertices of the coalition graph correspond one-to-one with the partition sets and two vertices are adjacent if and only if their corresponding sets form a coalition. In the paper [T. W. Haynes, J. T. Hedetniemi, S. T. Hedetniemi, A. A. McRae, R. Mohan, Discuss. Math. Graph Theory 43 (2023) 931–946], the authors proved that partition coalitions of cycles can generate only 27 coalition graphs and asked about the shortest cycle having the maximum number of coalition graphs. In this paper, we show that C 15 is the shortest graph having this property.