| Title: | Total ▫$k$▫-coalition: bounds, exact values and an application to double coalition |
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| Authors: | ID Brešar, Boštjan (Author)
ID Klavžar, Sandi (Author)
ID Samadi, Babak (Author) |
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| Files: |  PDF - Presentation file, download (480,42 KB)
MD5: D6904333362C0A52769727B3120247C1
 URL - Source URL, visit https://dmtcs.episciences.org/16036
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| 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 $\Omega=\{V_{1},\ldots,V_{|\Omega|}\}$ 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. |
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| Keywords: | total k-coalition, total k-domination, regular graph, double coalition |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | 18 str. |
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| Numbering: | Vol. 27, no. 3, article no. 2 |
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| PID: | 20.500.12556/DiRROS-23026  |
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| UDC: | 519.17 |
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| ISSN on article: | 1365-8050 |
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| DOI: | 10.46298/dmtcs.15231 |
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| COBISS.SI-ID: | 242867715 |
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| Note: |
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| Publication date in DiRROS: | 17.07.2025 |
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| Views: | 426 |
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| Downloads: | 222 |
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| Metadata: |    |
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