A characterization of graphs with given total coalition numbers

A set S of vertices in an isolate-free graph G is a total dominating set if every vertex of G is adjacent to some other vertex in S. A total coalition in G consists of two disjoint sets of vertices X and Y of G, neither of which is a total dominating set but whose union X∪Y is a total dominating set of G. Such sets X and Y are said to form a total coalition. A total coalition partition in G is a vertex partition Ψ={V₁,V₂,…,V_k} such that for all i∈[k], the set V_i forms a total coalition with another set V_j for some j, where j∈[k]∖{i}. The total coalition number C_t(G) in G equals the maximum order of a total coalition partition in G. It is known that if G is an isolate-free graph, then 2≤C_t(G)≤n. We characterize graphs with smallest possible total coalition number, that is, we characterize isolate-free graphs G satisfying C_t(G)=2. Moreover we characterize graphs G with δ(G)=1 satisfying C_t(G)=k for all k≥3.