2-reachable subsets in two-colored graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915341580369920 |
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| author | Gyarfas, Andras Sarkozy, Gabor N. |
| author_facet | Gyarfas, Andras Sarkozy, Gabor N. |
| contents | A subset $X$ of vertices in a graph $G$ is a {\em diameter 2 subset} if the distance of any two vertices of $X$ is at most two {\em in $G[X]$}. Relaxing this notion, a subset $X$ of vertices in a graph $G$ is a {\em 2-reachable subset} if the distance of any two vertices of $X$ is at most two {\em in $G$}. Related to recent attempts to strengthen a well-known conjecture of Ryser, English et al. conjectured that the vertices of a $2$-edge-colored cocktail party graph (the graph obtained from a complete graph with an even number of vertices by deleting a perfect matching) can be covered by the vertices of two monochromatic diameter $2$ subsets. In this note we prove the relaxed form of this conjecture, replacing diameter $2$ by $2$-reachable. An immediate corollary is that $2$-colored cocktail party graphs on $n$ vertices must contain a monochromatic $2$-reachable subset with at least $n\over 2$ vertices (and this is best possible). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11696 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 2-reachable subsets in two-colored graphs Gyarfas, Andras Sarkozy, Gabor N. Combinatorics A subset $X$ of vertices in a graph $G$ is a {\em diameter 2 subset} if the distance of any two vertices of $X$ is at most two {\em in $G[X]$}. Relaxing this notion, a subset $X$ of vertices in a graph $G$ is a {\em 2-reachable subset} if the distance of any two vertices of $X$ is at most two {\em in $G$}. Related to recent attempts to strengthen a well-known conjecture of Ryser, English et al. conjectured that the vertices of a $2$-edge-colored cocktail party graph (the graph obtained from a complete graph with an even number of vertices by deleting a perfect matching) can be covered by the vertices of two monochromatic diameter $2$ subsets. In this note we prove the relaxed form of this conjecture, replacing diameter $2$ by $2$-reachable. An immediate corollary is that $2$-colored cocktail party graphs on $n$ vertices must contain a monochromatic $2$-reachable subset with at least $n\over 2$ vertices (and this is best possible). |
| title | 2-reachable subsets in two-colored graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.11696 |