Edge-Maximal Graphs on Surfaces

TL;DR

This paper proves that for any surface with Euler genus g, edge-maximal graph embeddings are within a linear number of edges from a triangulation, solving a longstanding open problem.

Contribution

It establishes a linear bound on how close edge-maximal embeddings are to triangulations on surfaces, addressing an open problem from 1974.

Findings

Edge-maximal embeddings are within O(g) edges of a triangulation.

Provides the first linear bound on this difference for surfaces of genus g.

Solves a 50-year-old open problem in topological graph theory.

Abstract

We prove that for every surface $\Sigma$ of Euler genus $g$, every edge-maximal embedding of a graph in $\Sigma$ is at most $O(g)$ edges short of a triangulation of $\Sigma$. This provides the first answer to an open problem of Kainen (1974).