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

TL;DR

This paper provides a concise proof establishing the minimum edge density required to ensure two vertices of the same degree are connected by a fixed-length path, with a focus on asymptotic bounds.

Contribution

It offers a short, simplified proof for the density threshold guaranteeing such paths, addressing a question by Chen and Ma.

Findings

For large graphs, a density of at least 1/2+o(1) guarantees the path connection.

The bound is tight for paths of odd length.

The proof simplifies previous approaches to this problem.

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.