Edge-maximal graphs on orientable and some non-orientable surfaces

TL;DR

This paper investigates the structure of edge-maximal, non-complete graphs on various surfaces, identifying unique such graphs on some surfaces and constructing infinite families on others, revealing differences between orientable and non-orientable cases.

Contribution

It characterizes unique edge-maximal, non-triangulating graphs on certain surfaces and constructs infinite families on orientable surfaces of genus g, highlighting differences between surface types.

Findings

Unique such graphs on the projective plane, Klein bottle, and torus.

Infinite families of such graphs on orientable surfaces of genus g.

These graphs are close to, but do not form, triangulations.

Abstract

We study edge-maximal, non-complete graphs on surfaces that do not triangulate the surface. We prove that there is no such graph on the projective plane $\mathbb{N}_1$, $K_7-e$ is the unique such graph on the Klein bottle $\mathbb{N}_2$ and $K_8-E(C_5)$ is the unique such graph on the torus $\mathbb{S}_1$. In contrast to this for each $g\ge 2$ we construct an infinite family of such graphs on the orientable surface $\mathbb{S}_g$ of genus $g$, that are $\lfloor \frac{g}{2} \rfloor$ edges short of a triangulation.