Abstract
There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the $5$-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for $3$-edge-colorable cubic graphs, cubic graphs which are not $3$-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non $3$-edge-colorable graph is from being $3$-edge-colorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.
Highlights
We begin by commenting upon the main motivation of this paper
There are many hard conjectures in graph theory, like Tutte’s 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs
There are many hard problems in graph theory which can be solved in the general case if they can be solved for cubic graphs
Summary
Barcelona Graduate School of Mathematics and Departament de Matematiques Universitat Politecnica de Catalunya Jordi Girona 1-3, Modul C3, Campus Nord. Released under the CC BY-ND license (International 4.0)
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