On graph classes with minor-universal elements

TL;DR

This paper investigates the existence of universal graphs within various minor-closed classes, revealing conditions under which such universal elements exist or do not, and providing new insights into graph minors and their properties.

Contribution

It establishes the existence or non-existence of universal graphs in several natural classes, including minor-closed classes, and introduces new results on minor-closed classes and degree bounds.

Findings

Uncountably many minor-closed classes lack universal graphs.

Every $K_5$-minor-free graph is a minor of a degree-bounded $K_5$-minor-free graph.

Conditions for the existence of universal graphs in specific classes are characterized.

Abstract

A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-minor-free graph is a minor of a $K_5$-minor-free graph of maximum degree 22.