The shortest cycle having the maximal number of coalition graphs

A coalition in a graph G with a vertex set V consists of two disjoint sets V1, V2 ⊂ V , such that neither V1 nor V2 is a dominating set, but the union V1 ∪ V2 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.