Some snarks are worse than others
Edita M\'a\v{c}ajov\'a, Giuseppe Mazzuoccolo, Vahan Mkrtchyan, Jean, Paul Zerafa

TL;DR
This paper investigates specific subclasses of cubic graphs, called ${\
Contribution
It introduces parameters to further classify snarks, providing a natural scale to verify important conjectures in graph theory.
Findings
Decomposition of ${\cal S}_{\geq 5}$ into complexity-based subsets
New bounds on the number of perfect matchings needed
Relations established between parameters and longstanding conjectures
Abstract
Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic graph which is not 3--edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan-Raspaud Conjecture are examples of statements for which is crucial. In this paper, we study parameters which have the potential to further refine and thus enlarge the set of cubic graphs for which the mentioned conjectures can be…
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph theory and applications
