Connecting Coalition Partitions in the Cartesian Product of Graphs

For a graph G, the vertex partition $$\sigma =\{V_1, V_2,\cdots , V_k\}$$ is a connecting coalition partition if each $$V_i \in \sigma , |V_i| > 1$$ , induces a disconnected graph but a union with some $$V_j \in \sigma , |V_j| > 1$$ , $$V_i \cup V_j$$ induces a connected graph; otherwise $$V_i \in \sigma $$ is a singleton set containing a full vertex of G in $$\sigma $$ . Among all possible connecting coalition partitions for a graph G, the cardinality of the partition $$\sigma $$ having the maximum value is the connecting coalition number, $$\zeta (G)$$ . In this article, we determine the connecting coalition number for the Cartesian product of graphs. For graphs $$G_\mathcal {F}$$ , that do not attain a connecting coalition partition, the connecting coalition partition for $$G_\mathcal {F} \square G$$ , with a given graph G, is determined. The existence of the connecting coalition partition for the Cartesian product of any two non-trivial graphs is proved.