A generalization of Erd\H{o}s-Hajnal problem on paths with equal-degree endpoints

Abstract

Erd\H{o}s and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture.