On Universal Graphs for Trees and Tree-Like Graphs

TL;DR

The paper resolves long-standing questions about universal graphs for n-vertex trees and extends to treewidth-k graphs. It corrects a historic Chung–Graham error and achieves a new upper bound of $s(n)\le \frac{14}{5}n\log_2 n+O(n)$, while proving a general $O(kn\log n)$ edge bound for treewidth-k universality. The authors introduce the $h$-generated graphs $G_T^h$, develop a framework around DFS-ordered trees, and construct explicit trees $T_k$ to bound edges tightly; they also show interval-universality bounds, linking contiguous vertex sets to universal subgraphs. Together, these results advance the theory of universal graphs by giving near-tight bounds and a versatile constructive method grounded in treewidth and DFS-order techniques.

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 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 $O(kn\log n)$ edges that contains every $n$-vertex treewidth-$k$ graph as a subgraph. This is best possible in the sense that $Ω(kn\log{n})$ edges are required.

Figures (7)

• Figure 1: Types of edges in $\overrightarrow{G}^{h}_T$.
• Figure 2: (a) Defining $G_{2}$. (b) The $11$-vertex $G(3)$-admissible graph.
• Figure 3: Structures of $T'_{0}$ in Case I.
• Figure 4: Structures of $T'_{0}$ in Case II.
• Figure 5: Structures of $T'_{0}$ in Case III.

Theorems & Definitions (26)

• Theorem 1
• Theorem 2
• Definition 3
• Lemma 4
• proof
• Lemma 5
• proof
• Definition 6
• Theorem 7
• Definition 8