Abstract
A coalition in a graph \(G\) is a pair of disjoint nondominating subsets of its vertices \(V_1, V_2 \subset V(G)\) such that \(V_1\cup V_2\) is a dominating set. In the coalition partition \(\pi (G)=\{ V_1,V_2,\dots ,V_k \}\) , every nondominating set \(V_i\) is included in some coalition and if \(V_i\) is dominating, then it is a single-vertex set. A coalition partition of verticesof a graph \(G\) generates a coalition graph \(\text {CG}(G,\pi )\) whose vertices correspond to the partition sets, while two vertices areadjacent if the corresponding sets form a coalition. It is well known that all simple cycles of ordergreater than three generate in total 26 coalition graphs of order at most six. A universal cyclegenerates all such graphs. It is shown that only the cycles \(C_{3k}\) , \(k \ge 5\) , are universal.
