A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912517934022656 |
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| author | Liu, Zhen Zeng, Qinghou |
| author_facet | Liu, Zhen Zeng, Qinghou |
| contents | Answering a question of Erdős and Hajnal, Chen and Ma proved that for all \(n\geq600\) every graph with \(2n + 1\) vertices and at least \(n^2 + n+1\) edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph $K_{n,n+1}$ shows that this edge bound is sharp. In this paper, we develop a novel approach to handle graphs with large equal degrees, which enables us to establish the result for all $n\ge2$, thereby fully resolving the problem posed by Erdős and Hajnal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00523 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints Liu, Zhen Zeng, Qinghou Combinatorics Answering a question of Erdős and Hajnal, Chen and Ma proved that for all \(n\geq600\) every graph with \(2n + 1\) vertices and at least \(n^2 + n+1\) edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph $K_{n,n+1}$ shows that this edge bound is sharp. In this paper, we develop a novel approach to handle graphs with large equal degrees, which enables us to establish the result for all $n\ge2$, thereby fully resolving the problem posed by Erdős and Hajnal. |
| title | A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.00523 |