20.500.12556/RUL-170831 Total k-coalition: bounds, exact values and an application to double coalition 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. total k-coalition total k-domination regular graph double coalition celotna k-koalicija celotna k-dominacija regularni graf dvojna koalicija true false true Angleški jezik Slovenski jezik Članek v reviji 2025-07-17 11:06:04 2025-07-17 11:06:11 2025-07-20 03:41:45 0000-00-00 00:00:00 2025 0 0 18 str. no. 3, article no. 2 Vol. 27 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 1365-8050 10.46298/dmtcs.15231 242867715 2502.07310.pdf 2502.07310.pdf 1 85199769ABB46D2F8722F4716EB0AE85 b796909b0a97791bef284d9c326d9888bd3b7dbed56b394e785eaed62ca3822d ad147b91-62eb-11f0-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=213085 https://dmtcs.episciences.org/16036 1 56ece7ba-62eb-11f0-b232-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=213084 Fakulteta za matematiko in fiziko 0 0 0