On cubic graphs having the maximal coalition number
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917651021824000 |
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| author | Dobrynin, Andrey A. Golmohammadi, Hamidreza |
| author_facet | Dobrynin, Andrey A. Golmohammadi, Hamidreza |
| contents | A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is called a coalition partition $\mathcal{P}$ if every non-dominating set of $\mathcal{P}$ is a member of a coalition and every dominating set is a single-vertex set. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partition. It is known that for cubic graphs $C(G)\le 9$. The existence of cubic graphs with the maximal coalition number is an unsolved problem. In this paper, an infinite family of cubic graphs satisfying $C(G)=9$ is constructed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_06245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On cubic graphs having the maximal coalition number Dobrynin, Andrey A. Golmohammadi, Hamidreza Combinatorics A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is called a coalition partition $\mathcal{P}$ if every non-dominating set of $\mathcal{P}$ is a member of a coalition and every dominating set is a single-vertex set. The coalition number $C(G)$ of a graph $G$ is the maximum cardinality of its coalition partition. It is known that for cubic graphs $C(G)\le 9$. The existence of cubic graphs with the maximal coalition number is an unsolved problem. In this paper, an infinite family of cubic graphs satisfying $C(G)=9$ is constructed. |
| title | On cubic graphs having the maximal coalition number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.06245 |