On the size of universal graphs for spanning trees

Abstract

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.