Abstract
We define a total coalition in a graph G as a pair of disjoint subsets A 1,A 2 ⊆ A that satisfy the following conditions: (a) neither A 1 nor A 2 constitutes a total dominating set of G, and (b) A 1 ∪ A 2 constitutes a total dominating set of G. A total coalition partition of a graph G is a partition ϒ = {A 1,A 2, . ,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.
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