On Universal Graphs for Trees and Tree-Like Graphs
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Kaul, Neel Kim, Jaehoon Kim, Minseo Wood, David R. |
| author_facet | Kaul, Neel Kim, Jaehoon Kim, Minseo Wood, David R. |
| contents | Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and Tóth [Combin. Probab. Comput. 2023] discovered an error in the proof. By adding more edges to $G$ the error can be corrected, bringing the number of edges in $G$ to $\frac{7}{2}n \log_2 n + O(n). $
We make the first improvement to Chung and Graham's bound in over four decades by showing that there exists an $n$-vertex graph with $ \frac{14}{5}n \log_2 n + O(n) $ edges that contains every $n$-vertex tree as a subgraph.
Furthermore, we generalise this bound for treewidth-$k$ graphs by showing that there exists a graph with $O(kn\log(n/k+1))$ edges that contains every $n$-vertex treewidth-$k$ graph as a subgraph. This is best possible in the sense that $Ω(kn\log(n/k+1))$ edges are required. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Universal Graphs for Trees and Tree-Like Graphs Kaul, Neel Kim, Jaehoon Kim, Minseo Wood, David R. Combinatorics Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and Tóth [Combin. Probab. Comput. 2023] discovered an error in the proof. By adding more edges to $G$ the error can be corrected, bringing the number of edges in $G$ to $\frac{7}{2}n \log_2 n + O(n). $ We make the first improvement to Chung and Graham's bound in over four decades by showing that there exists an $n$-vertex graph with $ \frac{14}{5}n \log_2 n + O(n) $ edges that contains every $n$-vertex tree as a subgraph. Furthermore, we generalise this bound for treewidth-$k$ graphs by showing that there exists a graph with $O(kn\log(n/k+1))$ edges that contains every $n$-vertex treewidth-$k$ graph as a subgraph. This is best possible in the sense that $Ω(kn\log(n/k+1))$ edges are required. |
| title | On Universal Graphs for Trees and Tree-Like Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.22358 |