The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof
Combinatorics
2026-05-12 v1
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 enforces the existence of two such vertices. This bound is tight for paths of odd length.
Cite
@article{arxiv.2605.09798,
title = {The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof},
author = {Yamaan Attwa and Matías Azócar Carvajal and Simona Boyadzhiyska and Théo Pierron and Anusch Taraz},
journal= {arXiv preprint arXiv:2605.09798},
year = {2026}
}
Comments
5 pages, comments are welcome!