Journal of Combinatorics

Volume 7 (2016)

Number 2–3

Guest editors: Rong Luo and Cun-Quan Zhang

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

Pages: 453 – 479



Louis Esperet (Laboratoire G-SCOP, CNRS, Université Grenoble-Alpes, Grenoble, France)

Giuseppe Mazzuoccolo (Dipartimento di Informatica, Università degli Studi di Verona, Italy)

Michael Tarsi (The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel)


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 in [9], 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 $\phi_c (S) \geq 5$. 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.


snarks, circular flows, nowhere-zero flows, NP-completeness

2010 Mathematics Subject Classification


Published 23 February 2016