On the size of universal graphs for spanning trees

TL;DR

The paper corrects a long-standing flaw in the Chung–Graham bound for universal graphs that contain all $n$-vertex trees and then delivers a substantial improvement by using a carefully designed $(2,1)$-balanced tree $T$ and the square of its generated graph $G_T^2$, achieving $e(G)\le\frac{14}{3}n\log_3 n+O(n)$ (and thus $\le 2.945 n\log_2 n$). The authors develop a DFS-based $T$-admissible embedding framework with inductive lemmas ${\bf A_k},{\bf A_{k,t}}$, and construct an explicit tree $T_k$ to bound edge counts; they also introduce interval-universal graphs and the related $s^{int}(n)$ parameter, establishing upper and lower bounds and highlighting open questions. Overall, the work provides the first significant bound improvement in decades and deepens the understanding of universal graphs for spanning trees and interval-universal variants.

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.

Figures (4)

• Figure 1: A $G(k)$-admissible graph $G$
• Figure 2: Structures of $T'_{0}$
• Figure 3: Each tree $T"_i$ embeds as above. In addition, $v_t= \phi_1(w')$ and $v_{t-1}\in \{\phi_0(x_1), \phi_0(w)\}$.
• Figure 4: A few levels of $T_{k}$

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 2.1
• proof
• Definition 2.2
• Lemma 2.2
• proof
• Definition 3.1
• Lemma 3.1