Paths of length five with equal-degree endpoints

TL;DR

This paper extends previous results to show that for all sufficiently large n, the complete bipartite graph K_{n,n+1} uniquely avoids having two equal-degree vertices connected by a path of length five, confirming a conjecture for this case.

Contribution

It proves the uniqueness of K_{n,n+1} for avoiding length-five equal-degree paths in graphs with at least n^2+n edges, advancing the understanding of degree-path structures.

Findings

K_{n,n+1} is unique for avoiding length-five equal-degree paths for all n ≥ 11.

Confirms the next case of a conjecture on paths of odd length with equal-degree endpoints.

Extends previous results from length-three to length-five paths.

Abstract

Addressing a question posed by Erd\H{o}s 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 show that, for all $n \ge 11$, $K_{n,n+1}$ is the unique $(2n+1)$-vertex graph with at least $n^2+n$ edges that avoids two equal-degree vertices joined by a path of length five. This confirms the very next case of a general conjecture of Chen and Ma on paths of odd length with equal-degree endpoints.