Improved Universal Graphs for Trees

TL;DR

This paper improves bounds on the minimal number of edges in universal graphs for trees and graphs with bounded treewidth, using a novel embedding strategy based on separating trees into three parts.

Contribution

It introduces a new method for constructing universal graphs for trees and bounded treewidth graphs, reducing the upper bound on the number of edges needed.

Findings

Improved upper bound for universal graphs for trees to approximately 2.88 n log n.

Extended the method to graphs with bounded treewidth, establishing bounds involving w and n.

Developed a tree separation strategy based on ternary tree structures.

Abstract

A graph $G$ is universal for a class of graphs $\mathcal{C}$, if, up to isomorphism, $G$ contains every graph in $\mathcal{C}$ as a subgraph. In 1978, Chung and Graham asked for the minimal number $s(n)$ of edges in a graph with $n$ vertices that is universal for all trees with $n$ vertices. The currently best bounds assert that $n\ln n-O(n)\le s(n) \le C n\ln n+O(n)$, where $C = \frac{14}{5\ln 2} \approx 4.04$. We improve the upper bound to $c n\ln n + O(n)$, where $c = \frac{19}{6\ln 3} \approx 2.88$. In the proof we develop a strategy that, broadly speaking, is based on separating trees into three parts, thus enabling us to embed them in a structure that originates from ternary trees. Our method also applies to graphs with a bound on their treewidth. Let $s_w(n)$ be the minimum number of edges in a $n$-vertex graph that is universal for graphs with treewidth $w$. By performing a blow-up to our universal structure for trees we establish that $nw \ln(n/w) -O(nw) \leq s_w(n) \leq \frac{19}{6\ln3} n (w+1) \ln(n/w) + O(nw)$.