On the size of universal graphs for spanning trees
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915646215815168 |
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| author | Kim, Jaehoon Kim, Minseo |
| author_facet | Kim, Jaehoon Kim, Minseo |
| contents | Chung and Graham [J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most $\frac{5}{2}n \log_2 n + O(n)$ edges. We identify an error in their proof. This error can be corrected by adding more edges, which increases the number of edges to $e(G) \leq \frac{7}{2}n \log_2 n + O(n)$. Moreover, we further improve this by showing that there exists such an $n$-vertex graph with at most $\left(5- \frac{1}{3}\right)n \log_3 n + O(n) \leq 2.945 n \log_2 n$ edges. This is the first improvement of the bound since Chung and Graham's pioneering work four decades ago. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the size of universal graphs for spanning trees Kim, Jaehoon Kim, Minseo Combinatorics Chung and Graham [J. London Math. Soc., 1983] claimed that there exists an $n$-vertex graph $G$ containing all $n$-vertex trees as subgraphs that has at most $\frac{5}{2}n \log_2 n + O(n)$ edges. We identify an error in their proof. This error can be corrected by adding more edges, which increases the number of edges to $e(G) \leq \frac{7}{2}n \log_2 n + O(n)$. Moreover, we further improve this by showing that there exists such an $n$-vertex graph with at most $\left(5- \frac{1}{3}\right)n \log_3 n + O(n) \leq 2.945 n \log_2 n$ edges. This is the first improvement of the bound since Chung and Graham's pioneering work four decades ago. |
| title | On the size of universal graphs for spanning trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.19032 |