The structure of graphs with Circular flow number 5 or more, and the complexity of their recognition problem

TL;DR

This paper analyzes the structure of graphs with circular flow number 5 or more, introduces new construction methods for such snarks, and proves that recognizing these graphs is an NP-complete problem.

Contribution

It provides a detailed analysis of flow value sets, develops new methods for constructing snarks with high circular flow number, and establishes the NP-completeness of their recognition problem.

Findings

Sets of flow values are symmetric unions of open integer intervals in rac{1}{5}.

New construction methods generate a dense family of snarks with circular flow number rac{1}{5}.

Recognition of graphs with circular flow number rac{1}{5} NP-complete.

Abstract

For some time the Petersen graph has been the only known Snark with circular flow number $5$ (or more, as long as the assertion of Tutte's $5$-flow Conjecture is in doubt). Although infinitely many such snarks were presented eight years ago by Macajova and Raspaud, the variety of known methods to construct them and the structure of the obtained graphs were still rather limited. We start this article with an analysis of sets of flow values, which can be transferred through flow networks with the flow on each edge restricted to the open interval $(1,4)$ modulo $5$. All these sets are symmetric unions of open integer intervals in the ring $\mathbb{R}/5\mathbb{Z}$. We use the results to design an arsenal of methods for constructing snarks $S$ with circular flow number $\phi_c(S)\ge 5$. As one indication to the diversity and density of the obtained family of graphs, we show that it is sufficiently rich so that the corresponding recognition problem is NP-complete.