The Extremal Function for 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.

3 pages; the proof was modified to include the non-face triangles of the planar graph H; details were added for the cases k=1 and k=2