The extremal function for Petersen minors

TL;DR

This paper establishes the minimum edge threshold for graphs to contain a Petersen minor, characterizes Petersen-minor-free graphs near this threshold, and derives coloring and arboricity bounds for Petersen-minor-free graphs.

Contribution

It proves the exact extremal edge count for Petersen minors and characterizes all Petersen-minor-free graphs with edges close to this bound.

Findings

Graphs with at least 5n-8 edges contain a Petersen minor.

Petersen-minor-free graphs are 9-colorable.

Petersen-minor-free graphs have vertex arboricity at most 5.

Abstract

We prove that every graph with $n$ vertices and at least $5n-8$ edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$ edges. It follows that every graph containing no Petersen minor is 9-colourable and has vertex arboricity at most 5. These results are also best possible.