{
  "id": "2502.07310",
  "version": "v1",
  "published": "2025-02-11T07:15:52.000Z",
  "updated": "2025-02-11T07:15:52.000Z",
  "title": "Total $k$-coalition: bounds, exact values and an application to double coalition",
  "authors": [
    "Boštjan Brešar",
    "Sandi Klavžar",
    "Babak Samadi"
  ],
  "categories": [
    "math.CO"
  ],
  "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 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-11T07:15:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact values",
      "dominating set",
      "cubic graph",
      "application",
      "coalition partition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}