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

Brešar, Boštjan (author), Klavžar, Sandi (author), Samadi, Babak (author)

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 $\Omega=\{V_{1},\ldots,V_{|\Omega|}\}$ 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 concepts 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.

• Language: English
• Keywords: total k-coalition, total k-domination, regular graph, double coalition
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2025
• Year: 2025
• Number of pages: 18 str.
• Numbering: Vol. 27, no. 3, article no. 2
• PID: 20.500.12556/RUL-170831
• UDC: 519.17
• ISSN on article: 1365-8050
• DOI: 10.46298/dmtcs.15231
• COBISS.SI-ID: 242867715
• Publication date in RUL: 17.07.2025

Record is a part of a journal

• Title: Discrete mathematics & theoretical computer science
• Shortened title: Discret. math. theor. comput. sci.
• Publisher: DMTCS
• ISSN: 1365-8050
• COBISS.SI-ID: 8089433

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: celotna k-koalicija, celotna k-dominacija, regularni graf, dvojna koalicija