The Extremal Function for Apex Graphs

TL;DR

This paper provides a shorter, simpler proof for the extremal function related to apex graphs, confirming a conjecture about the maximum number of edges in linklessly embeddable graphs with given vertices and triangles.

Contribution

The paper offers a more concise proof of the extremal function for apex graphs, simplifying previous complex arguments.

Findings

Confirmed the conjecture for apex graphs.

Provided a shorter proof compared to previous work.

Strengthened understanding of extremal properties of apex graphs.

Abstract

McCarty and Thomas conjectured that a linklessly embeddable graph with $n\ge 7 $ vertices and $t$ triangles has at most $3n-9 +\frac{t}{3}$ edges. Thomas and Yoo proved this to be true for apex graphs. We give a shorter and simpler proof for the apex case.