Isometric universal graphs

TL;DR

This paper constructs small universal graphs containing all n-vertex graphs as isometric subgraphs, using a novel distance labelling scheme that could have broader applications.

Contribution

It introduces a new distance labelling scheme and demonstrates the existence of small universal graphs for all n-vertex graphs as isometric subgraphs.

Findings

Existence of universal graphs with 3^{n+O(log^2 n)} vertices

Development of a new distance labelling scheme

Potential broader applications of the labelling scheme

Abstract

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in $H$ coincide with the distances between the corresponding vertices in $G$. We show that for any integer $n\ge 1$, there is a graph on $3^{n+O(\log^2 n)}$ vertices that contains isometric copies of all $n$-vertex graphs. Our main tool is a new type of distance labelling scheme, whose study might be of independent interest.