The extremal functions for triangle-free graphs with excluded minors

TL;DR

This paper establishes upper bounds on the number of edges in certain triangle-free graphs with excluded minors, extending extremal graph theory results to graphs nearly planar or with specific minor restrictions.

Contribution

It introduces new extremal bounds for triangle-free graphs with specific minor exclusions, generalizing classical results to broader graph classes.

Findings

Graphs with a vertex removal leading to planarity have edges bounded by 3|V|-9+t/3.

Triangle-free graphs with no K_p-minor have edges bounded by (p-2)|V| - (p-2)^2 for p=2 to 9.

Results extend extremal graph theory to graphs with nearly planar structures and minor exclusions.

Abstract

We prove two results: 1. A graph $G$ on at least seven vertices with a vertex $v$ such that $G-v$ is planar and $t$ triangles satisfies $|E(G)| \leq 3|V(G)|- 9 + t/3$. 2. For $p=2,3,\ldots,9$, a triangle-free graph $G$ on at least $2p-5$ vertices with no $K_p$-minor satisfies $|E(G)|\leq (p-2)|V(G)| - (p-2)^2$.