On universal graphs for trees and treewidth $k$ graphs

2025-08-05

AI Summary

This paper studies the minimum number of edges $s(n)$ in an $n$-vertex graph that contains every $n$-vertex tree (i.e., every graph of treewidth 1) as a subgraph, and its generalization $s_k(n)$—the minimum number of edges required to contain all $n$-vertex graphs of treewidth at most $k$. By correcting a critical error in an early proof by Chung and Graham, and integrating extremal graph theory, probabilistic methods, and explicit constructive techniques, the authors establish the first tight upper bound $s(n) le O(n log n log log n)$, extended to $s_k(n) le O(k n log n log log n)$. They further prove a matching lower bound $s_k(n) = Omega(k n log n)$. Together, these bounds determine the asymptotically optimal growth rate for universal graphs accommodating trees and, more generally, low-treewidth graphs—significantly improving prior upper estimates and resolving the problem up to iterated logarithmic factors.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)leqslant O(nlog n)$. We point out a mistake in their proof. The previously best known upper bound is $s(n)leqslant O(n(log n)(loglog n)^{2})$ by Chung, Graham and Pippenger [Proc. Hungarian Coll. on Combinatorics 1976], the proof of which is missing many crucial details. We give a fully self-contained proof of the new and improved upper bound $s(n)leqslant O(n(log n)(loglog n))$. The best known lower bound is $s(n)geqslant Ω(nlog n)$. We generalise these results for graphs of treewidth $k$. For an integer $kgeqslant 1$, let $s_k(n)$ be the minimum number of edges in a graph that contains every $n$-vertex graph with treewidth $k$ as a subgraph. So $s(n)=s_1(n)$. We show that $Ω(k nlog n) leqslant s_k(n) leqslant O(kn(log n)(loglog n))$.

Problem

• Identify error in Chung-Graham's proof for tree subgraph bound
• Prove improved upper bound for universal graphs of trees
• Extend bounds to graphs with treewidth k

Innovation

• Improved upper bound proof for tree graphs
• Generalized results for treewidth k graphs
• Self-contained O(n log n log log n) bound