Morphology of small snarks

TL;DR

This paper classifies all snarks up to 36 vertices, revealing their structural composition and reasons for uncolourability, using computer-assisted analysis to understand their construction from fundamental components.

Contribution

It provides a comprehensive classification of small snarks, introduces a method for analyzing their structure, and generalizes findings to infinite families of snarks.

Findings

Most snarks are built from Petersen graph pieces

Snarks can be grouped into classes with common uncolourability reasons

Identified infinite families of cyclically 5-connected critical snarks

Abstract

The aim of this paper is to classify all snarks up to order $36$ and explain the reasons of their uncolourability. The crucial part of our approach is a computer-assisted structural analysis of cyclically $5$-connected critical snarks, which is justified by the fact that every other snark can be constructed from them by a series of simple operations while preserving uncolourability. Our results reveal that most of the analysed snarks are built up from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability. This sheds new light on the structure of all small snarks. Based on our analysis, we generalise certain individual snarks to infinite families and identify a rich family of cyclically $5$-connected critical snarks.