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arXiv · 2503.19569

A problem of Erdős and Hajnal on paths with equal-degree endpoints

Kaizhe Chen·Jie Ma
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Abstract

We address a problem posed by Erdős and Hajnal in 1991, proving that for all $n \geq 600$, every $(2n+1)$-vertex graph with at least $n^2 + n + 1$ edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph $K_{n,n+1}$ demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.

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Kaizhe Chen, Jie Ma. 2025-03-25. A problem of Erdős and Hajnal on paths with equal-degree endpoints. https://arxiv.org/abs/2503.19569

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