Cuts in matchings of 3-connected cubic graphs

TL;DR

This paper unifies several conjectures on Hamiltonicity, dichromatic number, and graph contractions within a common framework related to cuts in matchings, providing counterexamples and verifying conjectures for small cases.

Contribution

It introduces a unified framework for various conjectures on cubic and oriented graphs, finds a counterexample, and verifies a conjecture for small planar graphs.

Findings

Counterexample to Hochstättler's conjecture

Neumann-Lara conjecture verified for graphs up to 26 vertices

Unified framework for multiple graph conjectures

Abstract

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.