Bounded diameter variations of Ryser's conjecture
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arXiv
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| Format: | Preprint |
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2025
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| author | Gyarfas, Andras Sarkozy, Gabor N. |
| author_facet | Gyarfas, Andras Sarkozy, Gabor N. |
| contents | In this paper we study bounded diameter variations of the following form of Ryser's conjecture. For every graph $G=(V,E)$ with independence number $α(G)=α$ and integer $r\geq 2$, in every $r$-edge coloring of $G$ there is a cover of $V(G)$ by the vertices of $(r-1)α$ monochromatic connected components. Milićević initiated the question whether the diameters of the covering components can be bounded.
For any graph $G$ with $α(G)=2$ we show that in every 2-coloring of the edges, $V(G)$ can be covered by the vertices of two monochromatic subgraphs of diameter at most 4. This improves a result of DeBiasio et al., which in turn improved a result of Milićević. It remains open whether diameter $4$ can be strengthened to diameter $3$, we could do this only for certain graphs, including odd antiholes.
We propose also a somewhat orthogonal aspect of the problem. Suppose that we fix the diameter $d$ of the monochromatic components, how many do we need to cover the vertex set? For $d=2,2\le r \le 3$, the exact answer is $rα$ and for $d=4,r=2$, we prove the upper bound $\lfloor 3α/2\rfloor$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_02564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded diameter variations of Ryser's conjecture Gyarfas, Andras Sarkozy, Gabor N. Combinatorics In this paper we study bounded diameter variations of the following form of Ryser's conjecture. For every graph $G=(V,E)$ with independence number $α(G)=α$ and integer $r\geq 2$, in every $r$-edge coloring of $G$ there is a cover of $V(G)$ by the vertices of $(r-1)α$ monochromatic connected components. Milićević initiated the question whether the diameters of the covering components can be bounded. For any graph $G$ with $α(G)=2$ we show that in every 2-coloring of the edges, $V(G)$ can be covered by the vertices of two monochromatic subgraphs of diameter at most 4. This improves a result of DeBiasio et al., which in turn improved a result of Milićević. It remains open whether diameter $4$ can be strengthened to diameter $3$, we could do this only for certain graphs, including odd antiholes. We propose also a somewhat orthogonal aspect of the problem. Suppose that we fix the diameter $d$ of the monochromatic components, how many do we need to cover the vertex set? For $d=2,2\le r \le 3$, the exact answer is $rα$ and for $d=4,r=2$, we prove the upper bound $\lfloor 3α/2\rfloor$. |
| title | Bounded diameter variations of Ryser's conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.02564 |