Total Coalitions in Claw-Free Cubic Graphs Containing Double-Bonded Triangle-Units

Abstract 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 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 = \V₁, V₂, , Vₖ\ Ψ = V 1, V 2, …, V k such that for all i k i ∈ k, the set Vᵢ V i forms a total coalition with another set Vⱼ V j for some j, where j k \i\ j ∈ k \ i. We emphasize that none of the sets in Ψ is a total dominating set of G. The total coalition number Cₜ (G) C t (G) in G equals the maximum order of a total coalition partition in G. We study total coalitions in claw-free cubic graphs with certain structural properties, namely, graphs containing double-bonded triangle-units, that is, two vertex disjoint triangles joined by two edges.