Paths of length five with equal-degree endpoints
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914469588762624 |
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| author | Liu, Zhen Zeng, Qinghou |
| author_facet | Liu, Zhen Zeng, Qinghou |
| contents | Addressing a question posed by Erdős 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11664 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Paths of length five with equal-degree endpoints Liu, Zhen Zeng, Qinghou Combinatorics Addressing a question posed by Erdős 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. |
| title | Paths of length five with equal-degree endpoints |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.11664 |