Mutual-visibility problems in Kneser and Johnson graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910379458691072 |
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| author | Ekinci, Gülnaz Boruzanli Bujtás, Csilla |
| author_facet | Ekinci, Gülnaz Boruzanli Bujtás, Csilla |
| contents | Let $G$ be a connected graph and $\cal X \subseteq V(G)$. By definition, two vertices $u$ and $v$ are $\cal X$-visible in $G$ if there exists a shortest $u,v$-path with all internal vertices being outside of the set $\cal X$. The largest size of $\cal X$ such that any two vertices of $G$ (resp. any two vertices from $\cal X$) are $\cal X$-visible is the total mutual-visibility number (resp. the mutual-visibility number) of $G$.
In this paper, we determine the total mutual-visibility number of Kneser graphs, bipartite Kneser graphs, and Johnson graphs. The formulas proved for Kneser, and bipartite Kneser graphs are related to the size of transversal-critical uniform hypergraphs, while the total mutual-visibility number of Johnson graphs is equal to a hypergraph Turán number. Exact values or estimations for the mutual-visibility number over these graph classes are also established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mutual-visibility problems in Kneser and Johnson graphs Ekinci, Gülnaz Boruzanli Bujtás, Csilla Combinatorics 05C12, 05C38, 05C65, 05C76 Let $G$ be a connected graph and $\cal X \subseteq V(G)$. By definition, two vertices $u$ and $v$ are $\cal X$-visible in $G$ if there exists a shortest $u,v$-path with all internal vertices being outside of the set $\cal X$. The largest size of $\cal X$ such that any two vertices of $G$ (resp. any two vertices from $\cal X$) are $\cal X$-visible is the total mutual-visibility number (resp. the mutual-visibility number) of $G$. In this paper, we determine the total mutual-visibility number of Kneser graphs, bipartite Kneser graphs, and Johnson graphs. The formulas proved for Kneser, and bipartite Kneser graphs are related to the size of transversal-critical uniform hypergraphs, while the total mutual-visibility number of Johnson graphs is equal to a hypergraph Turán number. Exact values or estimations for the mutual-visibility number over these graph classes are also established. |
| title | Mutual-visibility problems in Kneser and Johnson graphs |
| topic | Combinatorics 05C12, 05C38, 05C65, 05C76 |
| url | https://arxiv.org/abs/2403.15645 |