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 $$X \cup Y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> </mml:math> 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 $$\Psi = \{V_1,V_2,\ldots ,V_k\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> such that for all $$i \in [k]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , the set $$V_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> forms a total coalition with another set $$V_j$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> for some j , where $$j \in [k] \setminus \{i\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>]</mml:mo> <mml:mo>\</mml:mo> <mml:mo>{</mml:mo> <mml:mi>i</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . We emphasize that none of the sets in $$\Psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ψ</mml:mi> </mml:math> is a total dominating set of G . The total coalition number $$C_t(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.