Year: 2025
Type: preprint
Abstract: We address a problem posed by Erd\H{o}s 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.
Source: arXiv (Cornell University)
Authors: Chen, Kaizhe, Ma, Jie
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6979277817
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