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arXiv · 2311.05500

Universality for graphs with bounded density

Noga Alon·Natalie Dodson·Carmen Jackson·Rose McCarty·Rajko Nenadov·Lani Southern
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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.

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Noga Alon, Natalie Dodson, Carmen Jackson, Rose McCarty, Rajko Nenadov, Lani Southern. 2024-01-11. Universality for graphs with bounded density. https://arxiv.org/abs/2311.05500

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