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 \(X \cup Y\) 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 \(\Psi = \{V_1,V_2,\ldots ,V_k\}\) such that for all \(i \in [k]\) , the set \(V_i\) forms a total coalition with another set \(V_j\) for some j, where \(j \in [k] \setminus \{i\}\) . We emphasize that none of the sets in \(\Psi \) is a total dominating set of G. The total coalition number \(C_t(G)\) 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.
