$k$-Fair Coalitions in Graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914036437745664 |
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| author | Jafari, Abbas Alikhani, Saeid |
| author_facet | Jafari, Abbas Alikhani, Saeid |
| contents | Let $G = (V,E)$ be a simple graph. A subset $S \subseteq V$ is called a $k$-fair dominating set if every vertex not in $S$ has exactly $k$ neighbors in $S$. Two disjoint sets $A, B \subseteq V$ form a $k$-fair coalition of $G$ if neither $A$ nor $B$ is a $k$-fair dominating set and the union $A \cup B$ is a $k$-fair dominating set of $G$. A partition $π= \{V_1, V_2, \ldots, V_m\}$ of $V$ is called a $k$-fair coalition partition, if every set $V_i\inπ$, either $V_i$ is a $k$-fair dominating set with exactly $k$ vertices, or $V_i$ is not a $k$-fair dominating set, but forms a $k$-fair coalition with some other set $V_j$ in $π$. The $k$-fair coalition number $C_{kf}(G)$ is the largest possible size of a $k$-fair coalition partition for $G$. The objective of this study is to initiate an examination into the notion of $k$-fair coalitions in graphs and present essential findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $k$-Fair Coalitions in Graphs Jafari, Abbas Alikhani, Saeid Combinatorics 05C69, 05C85 Let $G = (V,E)$ be a simple graph. A subset $S \subseteq V$ is called a $k$-fair dominating set if every vertex not in $S$ has exactly $k$ neighbors in $S$. Two disjoint sets $A, B \subseteq V$ form a $k$-fair coalition of $G$ if neither $A$ nor $B$ is a $k$-fair dominating set and the union $A \cup B$ is a $k$-fair dominating set of $G$. A partition $π= \{V_1, V_2, \ldots, V_m\}$ of $V$ is called a $k$-fair coalition partition, if every set $V_i\inπ$, either $V_i$ is a $k$-fair dominating set with exactly $k$ vertices, or $V_i$ is not a $k$-fair dominating set, but forms a $k$-fair coalition with some other set $V_j$ in $π$. The $k$-fair coalition number $C_{kf}(G)$ is the largest possible size of a $k$-fair coalition partition for $G$. The objective of this study is to initiate an examination into the notion of $k$-fair coalitions in graphs and present essential findings. |
| title | $k$-Fair Coalitions in Graphs |
| topic | Combinatorics 05C69, 05C85 |
| url | https://arxiv.org/abs/2509.11358 |