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

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 Ψ = ₁,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] \ j ∈ [ k ] \ { i } . We emphasize that none of the sets in Ψ 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.