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arXiv · 2605.09798

The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof

Yamaan Attwa·Matías Azócar Carvajal·Simona Boyadzhiyska·Théo Pierron·Anusch Taraz
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Abstract

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.

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Yamaan Attwa, Matías Azócar Carvajal, Simona Boyadzhiyska, Théo Pierron, Anusch Taraz. 2026-05-10. The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof. https://arxiv.org/abs/2605.09798

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