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) \le \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 \cdot n\log_2(n)) edges. This is the first improvement of the bound since Chung and Graham's pioneering work four decades ago.