Paths of even length with equal-degree endpoints

Abstract

Addressing a question posed by Erdős and Hajnal, Chen and Ma proved that, for all $n \ge 600$, the complete bipartite graph $K_{n,n+1}$ is the unique graph on $2n+1$ vertices with at least $n^2+n$ edges that contains no two vertices of equal degree joined by a path of length three. In this paper, we extend this result and prove that for every fixed integer \(\ell\ge 2\) and sufficiently large \(n\), the unique \(2n\)-vertex graph with at least \((n^2+n)/2\) edges that contains no two vertices of equal degree joined by a path of length \(2\ell\) is the half graph \(H_n\). This resolves the problem posed by Chen and Ma, as well as a related question of Attwa, Azócar Carvajal, Boyadzhiyska, Pierron, and Taraz concerning paths of even length with equal-degree endpoints.