Paths of even length with equal-degree endpoints

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 $\mathrm{\ell }\ge 2$ and sufficiently large $n$, the unique $2n$-vertex graph with at least $(n^2+n)\mathrm{/}2$ edges that contains no two vertices of equal degree joined by a path of length $2\mathrm{\ell }$ 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.