Computational results and new bounds for the circular flow number of snarks

• Published in: Discrete mathematics Vol. 343; no. 10; p. 112026
• Main Authors: Goedgebeur, Jan, Mattiolo, Davide, Mazzuoccolo, Giuseppe
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2020
• Subjects: Algorithm; Bisection; Circular flow; Cubic graph; Dot product; Snark; Bisection; Snark; Circular flow; Dot product; Algorithm; Cubic graph
• ISSN: 0012-365X; 1872-681X; 1872-681X
• DOI: 10.1016/j.disc.2020.112026

It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computation of the circular flow number of a bridgeless cubic graph. Using this procedure, we determine the circular flow number of all snarks on up to 36 vertices as well as the circular flow number of various famous snarks. After that, as combination of the use of this algorithm with new theoretical results, we present an infinite family of snarks of order 8k+2 whose circular flow numbers meet a general lower bound presented by Lukot’ka and Škoviera in 2008. In particular this answers a question proposed in their paper. Moreover, we improve the best known upper bound for the circular flow number of Goldberg snarks and we conjecture that this new upper bound is optimal.