On Universal Graphs for Trees and Tree-Like Graphs

TL;DR

This paper improves the upper bound on the number of edges needed in a universal graph for all n-vertex trees, and generalizes the result to graphs of bounded treewidth, matching the lower bound.

Contribution

It provides the first improvement to Chung and Graham's bound in over four decades and extends the result to treewidth-k graphs with optimal edge bounds.

Findings

Constructed n-vertex graphs with fewer edges that contain all n-vertex trees.

Generalized bounds to treewidth-k graphs with tight asymptotic match.

Improved the known upper bound from 7/2 n log n to 14/5 n log n edges.

Abstract

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 T\'oth [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 $\Omega(kn\log(n/k+1))$ edges are required.