Snarks without Small Cycles

Abstract

Snarks are nontrivial cubic graphs whose edges cannot be colored with three colors. Jaeger and Swart conjectured that any snark has girth (the length of the shortest cycle) at most 6. This problem is also known as thegirth conjectureof snarks. The aim of this paper is to give a negative solution of this conjecture and construct snarks with arbitrarily large girths. For instance, if we use known constructions of cubic graphs with large girths, then we can explicitly construct cyclically 5-edge-connected snarks of ordernand with girth at least[formula].