Total $k$-coalition: bounds, exact values and an application to double coalition

Abstract

Let $G=\big{(}V(G),E(G)\big{)}$ be a graph with minimum degree $k$. A subset $S\subseteq V(G)$ is called a total $k$-dominating set if every vertex in $G$ has at least $k$ neighbors in $S$. Two disjoint sets $A,B\subset V(G)$ form a total $k$-coalition in $G$ if none of them is a total $k$-dominating set in $G$ but their union $A\cup B$ is a total $k$-dominating set. A vertex partition $\hat{I}\text{c◯}=\{V_1,\dots ,V_{|\hat{I}\text{c◯}|}\}$ of $G$ is a total $k$-coalition partition if each set $V_i$ forms a total $k$-coalition with another set $V_j$. The total $k$-coalition number ${\mathrm{TC}}_k(G)$ of $G$ equals the maximum cardinality of a total $k$-coalition partition of $G$. In this paper, the above-mentioned concept are investigated from combinatorial points of view. Several sharp lower and upper bounds on ${\mathrm{TC}}_k(G)$ are proved, where the main emphasis is given on the invariant when $k=2$. As a consequence, the exact values of ${\mathrm{TC}}_2(G)$ when $G$ is a cubic graph or a $4$-regular graph are obtained. By using similar methods, an open question posed by Henning and Mojdeh regarding double coalition is answered. Moreover, ${\mathrm{TC}}_3(G)$ is determined when $G$ is a cubic graph.