A generalization of Erdős-Hajnal problem on paths with equal-degree endpoints
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914531349889024 |
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| author | Zhao, Xiamiao Wang, Yichen Lu, Mei |
| author_facet | Zhao, Xiamiao Wang, Yichen Lu, Mei |
| contents | Erdős and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03825 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A generalization of Erdős-Hajnal problem on paths with equal-degree endpoints Zhao, Xiamiao Wang, Yichen Lu, Mei Combinatorics Erdős and Hajnal proposed a problem that: is it true that every $(2n+1)$-vertex graph with $n^2+n+1$ edges contains two vertices of equal degree connected by a path of length three? The edge bound is sharp by the complete bipartite graph $K_{n,n+1}$. Recently, Chen and Ma [Journal of Combinatorial Theory, Series B, 179:1-18, 2026] answered this problem affirmatively for every $n \ge 600$. In the same paper, they further conjectured that for sufficiently large $n$, the statement is true if we replace the path of length three by a path of fixed odd length. In this paper, we confirm their conjecture. |
| title | A generalization of Erdős-Hajnal problem on paths with equal-degree endpoints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.03825 |