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

TL;DR

We address Erdős-Hajnal's problem on edge-density forcing two vertices of equal degree connected by a path of length three in $(2n+1)$-vertex graphs with $e$ edges. Our approach combines degree-structure lemmas, upper-lower bounds on the maximum degree $\Delta$, Mantel-type enhancements, and triangle-supersaturation to pin down extremal configurations. We prove the unique extremal graph is $K_{n,n+1}$ for odd order with $e\ge n^2+n$, and the analogous unique extremal for even order is $K_{n-1,n+1}$ with $e\ge n^2-1$, thus resolving the problem and its sharpness. We also define the extremal function $p_\ell(n)$ for paths of length $\ell$ and present tight results for $p_3$ on odd and even orders, plus bounds and open questions for $p_1$ and $p_2$ and longer paths.

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.

Theorems & Definitions (22)

• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof