Strong embeddings of minimum genus

TL;DR

This paper constructs counterexamples to a folklore conjecture by showing certain cubic graphs can only be embedded on very high genus surfaces, challenging the idea that all bridgeless cubic graphs can be embedded on their own genus surface.

Contribution

It provides the first known counterexamples to the conjecture that all bridgeless cubic graphs can be embedded on their own genus surface with face boundaries as cycles.

Findings

Certain cubic graphs require very high genus surfaces for closed 2-cell embeddings

Counterexamples challenge the folklore conjecture about embeddings of bridgeless cubic graphs

Main results may be of independent interest for graph embedding theory

Abstract

A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.