Universality for graphs with bounded density

TL;DR

This work studies universal graphs for the family $\mathcal{H}_d(n)$ of $n$-vertex graphs with density at most $d>1$. It develops a unifying product-construction framework augmented with matroid decomposition, expanders, discrepancy theory, and random-walk techniques to achieve sparse $\mathcal{H}_d(n)$-universal graphs. The main results show: (i) for any rational $d>1$, there exists a universal graph with $e(G) \le C(d) n^{2-1/(\lceil d \rceil+1)}$ edges; (ii) when $d$ is an integer and maximum degree is bounded, the bound improves to near-optimal or optimal $e(G) \le C(D,d) n^{2-1/d}$; and (iii) stronger variants with a small blow-up factor are obtained for the non-integer rational case. These results unify and extend prior bounds for sparse universal graphs and contribute new algorithmic embedding methods, with potential implications in graph synthesis and related computational tasks. Key techniques include a product/blow-up construction, a matroid-based decomposition to handle unicyclic components, explicit expanders for embedding, and randomized tree-homomorphisms to control dispersion during embeddings, all culminating in a coherent framework for universality in bounded-density graph families.$

Abstract

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $Ω(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction.

Theorems & Definitions (28)

• Conjecture 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• proof : Proof of Proposition \ref{['prop:lower_bound']}
• Lemma 3.1
• proof
• Lemma 3.2: alon02treecut