Lower Bounds for Induced-Universal Graphs

TL;DR

This paper establishes new lower bounds on the size of induced-universal graphs for various graph families, highlighting the complexity and limitations in constructing such graphs.

Contribution

It provides the first known lower bounds for induced-universal graphs for several graph families, including planar and minor-closed graphs, and introduces methods linking colorings and decompositions.

Findings

Induced-universal graph for n-vertex planar graphs must have at least 10.52n vertices.

Any family of fewer than 137 planar graphs requires larger induced-universal graphs.

Results extend to other graph families like trees, outerplanar, and series-parallel graphs.

Abstract

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs must have at least $10.52n$ vertices. We also show that the number of conflicting graphs to consider in order to beat this lower bound is at least $137$. In other words, any family of less than $137$ planar graphs of $n$ vertices has an induced-universal graph with less than $10.52n$ vertices, stressing the difficulty in beating such lower bounds. Similar results are developed for other graph families, including but not limited to, trees, outerplanar graphs, series-parallel graphs, $K_{3,3}$-minor free graphs. As a byproduct, we show that any family of $t$ graphs of $n$ vertices having small chromatic number and sublinear pathwidth, like any proper minor-closed family, has an induced-universal graph with less than $\frac{15}{7} \sqrt{t} \cdot n$ vertices. This is achieved by making a bridge between equitable colorings, combinatorial designs, and path-decompositions.