Spanning trees for many different numbers of leaves
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916487474708480 |
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| author | Noguchi, Kenta Zamfirescu, Carol T. |
| author_facet | Noguchi, Kenta Zamfirescu, Carol T. |
| contents | Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13674 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spanning trees for many different numbers of leaves Noguchi, Kenta Zamfirescu, Carol T. Combinatorics 05C05 (Primary) 05C10 (Secondary) Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves. |
| title | Spanning trees for many different numbers of leaves |
| topic | Combinatorics 05C05 (Primary) 05C10 (Secondary) |
| url | https://arxiv.org/abs/2312.13674 |