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AbstractWe define a total coalition in a graph G as a pair of disjoint subsets A1, A2 ⊆ A that satisfy the following conditions: (a) neither A1 nor A2 constitutes a total dominating set of G, and (b) A1 ∪ A2 constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = A1, A2,. . . , Ak of its vertex set such that no subset of ϒ acts as a total dominating set of G, but for every set Ai ∈ ϒ, there exists a set Aj ∈ ϒ such that Ai and Aj combine to form a total coalition. We define the total coalition number of G as the maximum cardinality of a total coalition partition of G, and we denote it by Ct (G). The purpose of this paper is to begin an investigation into the characteristics of total coalition in graphs. Mathematics Subject Classification (2020): 05C69Key words: Total dominating setcoalitiontotal coalitiontree
Alikhani et al. (Sun,) studied this question.