On the connected coalition number of graphs

For a graph \(G=(V,E)\), a pair of vertex disjoint sets \(A₁\) and \(A₂\) form a connected coalition of \(G\), if \(A₁∪ A₂\) is a connected dominating set, but neither \(A₁\) nor \(A₂\) is a connected dominating set. A connected coalition partition of \(G\) is a partition \(Φ\) of \(V(G)\) such that each set in \(Φ\) either consists of only a singe vertex with the degree \( V(G)-1\), or forms a connected coalition of \(G\) with another set in \(Φ\). The connected coalition number of \(G\), denoted by \(CC(G)\), is the largest possible size of a connected coalition partition of \(G\). In this paper, we characterize graphs that satisfy \(CC(G)=2\). Moreover, we obtain the connected coalition number for unicycle graphs and for the corona product and join of two graphs. Finally, we give a lower bound on the connected coalition number of the Cartesian product and the lexicographic product of two graphs.