Algorithms for the Generation of Snarks

Abstract

The essential requirement for a cubic graph to be called a snark is that it can not be edge-coloured with three colours. To avoid trivial cases, varying restrictions on the connectivity are imposed. Snarks are not only interesting in themselves, but also a valuable test field for conjectures about graphs that are not snarks and sometimes not even cubic. For many important open problems in graph theory it is proven that minimal counterexamples would be snarks. We give two new algorithms for the generation of snarks and results of computer programs implementing these algorithms. One algorithm is for snarks with girth exactly 4 and is used for generating complete lists of girth 4 snarks on up to 40 vertices. The second algorithm lists snarks with girth at least 5 and is used for generating complete lists of such snarks on up to 38 vertices. We also give complete lists of strong snarks (in the terminology of Jaeger) on up to 40 vertices.

Figures (8)

• Figure 1: Construction and reduction operations used for girth 4 snarks. The construction operation applied to the path $e_1,e_2,e_3$ is denoted as $C(\{e_1,e_2,e_3\})$. The reduction operation applied to the pair $e'_1,e'_2$ of edges that are opposite in a 4-cycle, is denoted as $R(\{e'_1,e'_2\})$.
• Figure 2: The colours of the edges adjacent to the 4-cycle in the proof of Lemma \ref{['lem:deco']}.
• Figure 3: The notation for vertices and edges for the proof of Lemma \ref{['lem:reduce']}.
• Figure 4: An illustration of an argument in the proof of Lemma \ref{['lem:double']}.
• Figure 5: A legal labelling for cycles with lengths $5,7,4,6$.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof