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arXiv · 2604.11664

Paths of length five with equal-degree endpoints

Zhen Liu·Qinghou Zeng
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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.

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Zhen Liu, Qinghou Zeng. 2026-04-13. Paths of length five with equal-degree endpoints. https://arxiv.org/abs/2604.11664

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