The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911669996748800 |
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| author | Attwa, Yamaan Carvajal, Matías Azócar Boyadzhiyska, Simona Pierron, Théo Taraz, Anusch |
| author_facet | Attwa, Yamaan Carvajal, Matías Azócar Boyadzhiyska, Simona Pierron, Théo Taraz, Anusch |
| contents | Addressing a question posed by Chen and Ma from an asymptotic point of view, we present a short proof for the edge density needed to guarantee that two vertices of the same degree are connected by a path of a fixed length. In particular, we show that for any sufficiently large graph, a density of at least $1/2+o(1)$ enforces the existence of two such vertices. This bound is tight for paths of odd length. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09798 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof Attwa, Yamaan Carvajal, Matías Azócar Boyadzhiyska, Simona Pierron, Théo Taraz, Anusch Combinatorics 05C35, 05C38 Addressing a question posed by Chen and Ma from an asymptotic point of view, we present a short proof for the edge density needed to guarantee that two vertices of the same degree are connected by a path of a fixed length. In particular, we show that for any sufficiently large graph, a density of at least $1/2+o(1)$ enforces the existence of two such vertices. This bound is tight for paths of odd length. |
| title | The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof |
| topic | Combinatorics 05C35, 05C38 |
| url | https://arxiv.org/abs/2605.09798 |