An algorithm and new bounds for the circular flow number of snarks

TL;DR

This paper introduces a new algorithm for computing the circular flow number of bridgeless cubic graphs, applies it to snarks up to 36 vertices, and establishes new bounds and infinite families related to this parameter.

Contribution

It presents a more efficient algorithm for calculating the circular flow number and combines it with theoretical results to identify new bounds and infinite families of snarks with specific flow properties.

Findings

Determined circular flow numbers for all snarks up to 36 vertices.

Constructed an infinite family of snarks meeting a new lower bound.

Improved the upper bound for Goldberg snarks' circular flow number.

Abstract

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 design a more efficient algorithm for the computation of the circular flow number of a bridgeless cubic graph. Using this algorithm, we determine the circular flow number of all snarks up to 36 vertices as well as the circular flow number of various famous snarks. After that, as combination of the use of our 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 Skoviera 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. Finally, we discuss a possible extension to the computation of the circular flow number in the non-regular case.