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

TL;DR

This paper fully resolves a problem posed by Erd ext{"o}s and Hajnal regarding the existence of paths of length three between vertices of equal degree in graphs with a specific number of vertices and edges, extending previous results to all n ≥ 2.

Contribution

The paper introduces a new approach to analyze graphs with large equal degrees, enabling the extension of the Erd ext{"o}s-Hajnal problem solution to all n ≥ 2.

Findings

The result holds for all n ≥ 2, not just n ≥ 600.

The edge bound n^2 + n + 1 is sharp, as shown by the complete bipartite graph K_{n,n+1}.

A novel method for handling graphs with large equal degrees is developed.

Abstract

Answering a question of Erd\H{o}s and Hajnal, Chen and Ma proved that for all \(n\geq600\) every graph with \(2n + 1\) vertices and at least \(n^2 + n+1\) edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph $K_{n,n+1}$ shows that this edge bound is sharp. In this paper, we develop a novel approach to handle graphs with large equal degrees, which enables us to establish the result for all $n\ge2$, thereby fully resolving the problem posed by Erd\H{o}s and Hajnal.