Total Coalitions in Claw-Free Cubic Graphs Containing Double-Bonded Triangle-Units
Zoltán L. Blázsik, Michael A. Henning, Shahin Nosrat Jogan · Graphs and Combinatorics · 2026
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 $$X \cup Y$$ 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 $$\Psi = \{V_1,V_2,\ldots ,V_k\}$$ Ψ = { V 1 , V 2 , … , V k } such that for all $$i \in [k]$$ i ∈ [ k ] , the set $$V_i$$ V i forms a total coalition with another set $$V_j$$ V j for some j , where $$j \in [k] \setminus \{i\}$$ j ∈ [ k ] \ { i } . We emphasize that none of the sets in $$\Psi $$ Ψ is a total dominating set of G . The total coalition number $$C_t(G)$$ 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.