Maximum planar subgraphs in dense graphs

TL;DR

This paper determines the optimal constant for the size of large planar subgraphs in dense graphs with minimum degree less than half the vertices, refining previous bounds and extending understanding of graph planarity in dense settings.

Contribution

It finds the exact value of the constant C for all gamma<1/2 in dense graphs, improving upon prior bounds and characterizing maximum planar subgraphs.

Findings

Optimal C value identified for gamma<1/2

Maximum planar subgraphs contain at least 2n - C edges

Results apply to sufficiently large graphs

Abstract

K\"uhn, Osthus and Taraz showed that for each \gamma>0 there exists C such that any n-vertex graph with minimum degree \gamma n contains a planar subgraph with at least 2n-C edges. We find the optimum value of C for all \gamma<1/2 and sufficiently large n.