A unified approach to construct snarks with circular flow number 5

Perfect Matching Index versus Circular Flow Number of a Cubic Graph

SIAM Journal on Discrete Mathematics ◽

10.1137/20m1359407 ◽

2021 ◽

Vol 35 (2) ◽

pp. 1287-1297

Author(s):

Edita Máčajová ◽

Martin Škoviera

Keyword(s):

Perfect Matching ◽

Cubic Graph ◽

Flow Number ◽

Circular Flow ◽

Matching Index

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Circular flow number of generalized Blanuša snarks

Discrete Mathematics ◽

10.1016/j.disc.2013.01.010 ◽

2013 ◽

Vol 313 (8) ◽

pp. 975-981 ◽

Cited By ~ 1

Author(s):

Robert Lukot’ka

Keyword(s):

Flow Number ◽

Circular Flow

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

Journal of Combinatorics ◽

10.4310/joc.2016.v7.n2.a12 ◽

2016 ◽

Vol 7 (2–3) ◽

pp. 453-479

Author(s):

Louis Esperet ◽

Giuseppe Mazzuoccolo ◽

Michael Tarsi

Keyword(s):

Recognition Problem ◽

Flow Number ◽

Circular Flow

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The Circular Flow Number of a 6-Edge Connected Graph is Less Than Four

COMBINATORICA ◽

10.1007/s004930200025 ◽

2002 ◽

Vol 22 (3) ◽

pp. 455-459 ◽

Cited By ~ 6

Author(s):

Anna Galluccio ◽

Luis A. Goddyn

Keyword(s):

Connected Graph ◽

Flow Number ◽

Circular Flow

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Treelike Snarks

The Electronic Journal of Combinatorics ◽

10.37236/6008 ◽

2016 ◽

Vol 23 (3) ◽

Cited By ~ 4

Author(s):

Marién Abreu ◽

Tomáš Kaiser ◽

Domenico Labbate ◽

Giuseppe Mazzuoccolo

Keyword(s):

Infinite Family ◽

Double Cover ◽

Perfect Matchings ◽

Computer Assisted ◽

Flow Number ◽

Circular Flow

We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number $\phi_C (G)\ge5$ and admit a 5-cycle double cover.

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Determining the Circular Flow Number of a Cubic Graph

The Electronic Journal of Combinatorics ◽

10.37236/9607 ◽

2021 ◽

Vol 28 (1) ◽

Author(s):

Robert Lukoťka

Keyword(s):

Polynomial Algorithms ◽

Cubic Graphs ◽

Cubic Graph ◽

Flow Number ◽

Circular Flow ◽

Bridgeless Graph

A circular nowhere-zero $r$-flow on a bridgeless graph $G$ is an orientation of the edges and an assignment of real values from $[1, r-1]$ to the edges in such a way that the sum of incoming values equals the sum of outgoing values for every vertex. The circular flow number, $\phi_c(G)$, of $G$ is the infimum over all values $r$ such that $G$ admits a nowhere-zero $r$-flow. A flow has its underlying orientation. If we subtract the number of incoming and the number of outgoing edges for each vertex, we get a mapping $V(G) \to \mathbb{Z}$, which is its underlying balanced valuation. In this paper we describe efficient and practical polynomial algorithms to turn balanced valuations and orientations into circular nowhere zero $r$-flows they underlie with minimal $r$. Using this algorithm one can determine the circular flow number of a graph by enumerating balanced valuations. For cubic graphs we present an algorithm that determines $\phi_c(G)$ in case that $\phi_c(G) \leqslant 5$ in time $O(2^{0.6\cdot|V(G)|})$. If $\phi_c(G) > 5$, then the algorithm determines that $\phi_c(G) > 5$ and thus the graph is a counterexample to Tutte's $5$-flow conjecture. The key part is a procedure that generates all (not necessarily proper) $2$-vertex-colourings without a monochromatic path on three vertices in $O(2^{0.6\cdot|V(G)|})$ time. We also prove that there is at most $2^{0.6\cdot|V(G)|}$ of them.

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