A problem of Erdős and Hajnal on paths with equal-degree endpoints
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913757282697216 |
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| author | Chen, Kaizhe Ma, Jie |
| author_facet | Chen, Kaizhe Ma, Jie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A problem of Erdős and Hajnal on paths with equal-degree endpoints Chen, Kaizhe Ma, Jie Combinatorics 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. |
| title | A problem of Erdős and Hajnal on paths with equal-degree endpoints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.19569 |