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

• Published in: Discrete Mathematics and Theoretical Computer Science Vol. 27:3; no. Graph Theory; pp. 1 - 18
• Main Authors: Brešar, Boštjan, Klavžar, Sandi, Samadi, Babak
• Format: Journal Article
• Language: English
• Published: Nancy DMTCS 01.10.2025; Discrete Mathematics & Theoretical Computer Science
• Subjects: combinatorics; Fuzzy sets; Graph theory; Graphs; Mathematical research; Mathematics; Set theory; Slovenia
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.15231

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 $Ω=\{V_{1},\ldots,V_{|Ω|}\}$ 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 ${\rm TC}_{k}(G)$ 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 ${\rm TC}_{k}(G)$ are proved, where the main emphasis is given on the invariant when $k=2$. As a consequence, the exact values of ${\rm TC}_2(G)$ 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, ${\rm TC}_3(G)$ is determined when $G$ is a cubic graph.