Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.22977 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908736731217920 |
|---|---|
| author | Sun, Wensheng Yang, Yujun Xu, Shou-Jun |
| author_facet | Sun, Wensheng Yang, Yujun Xu, Shou-Jun |
| contents | A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22977 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes Sun, Wensheng Yang, Yujun Xu, Shou-Jun Combinatorics A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching. |
| title | A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.22977 |