Perfect Matching Index versus Circular Flow Number of a Cubic 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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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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Even Cycles and Even 2-Factors in the Line Graph of a Simple Graph

The Electronic Journal of Combinatorics ◽

10.37236/5660 ◽

2017 ◽

Vol 24 (4) ◽

Cited By ~ 1

Author(s):

Arrigo Bonisoli ◽

Simona Bonvicini

Keyword(s):

Perfect Matching ◽

Line Graph ◽

Simple Graph ◽

Connected Components ◽

Regular Graphs ◽

Necessary And Sufficient Condition ◽

Cubic Graph ◽

Cycle Decomposition ◽

Odd Degree ◽

Necessary And Sufficient

Let $G$ be a connected graph with an even number of edges. We show that if the subgraph of $G$ induced by the vertices of odd degree has a perfect matching, then the line graph of $G$ has a $2$-factor whose connected components are cycles of even length (an even $2$-factor). For a cubic graph $G$, we also give a necessary and sufficient condition so that the corresponding line graph $L(G)$ has an even cycle decomposition of index $3$, i.e., the edge-set of $L(G)$ can be partitioned into three $2$-regular subgraphs whose connected components are cycles of even length. The more general problem of the existence of even cycle decompositions of index $m$ in $2d$-regular graphs is also addressed.

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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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Perfect Matching Covers of Cubic Graphs of Oddness 2

The Electronic Journal of Combinatorics ◽

10.37236/7175 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Wuyang Sun ◽

Fan Wang

Keyword(s):

Perfect Matching ◽

Perfect Matchings ◽

Cubic Graphs ◽

Cubic Graph

A perfect matching cover of a graph $G$ is a set of perfect matchings of $G$ such that each edge of $G$ is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer $c$ does exist such that every bridgeless cubic graph has a perfect matching cover of order at most $c$. In this paper, we show that a bridgeless cubic graph $G$ has a perfect matching cover of order at most 11 if $G$ has a 2-factor in which the number of odd circuits is 2.

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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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Flows in Signed Graphs with Two Negative Edges

The Electronic Journal of Combinatorics ◽

10.37236/4458 ◽

2018 ◽

Vol 25 (2) ◽

Cited By ~ 1

Author(s):

Edita Rollová ◽

Michael Schubert ◽

Eckhard Steffen

Keyword(s):

Signed Graph ◽

Cubic Graphs ◽

Cubic Graph ◽

Signed Graphs ◽

Flow Number ◽

Edge Connectivity ◽

Cyclic Edge Connectivity

The presented paper studies the flow number $F(G,\sigma)$ of flow-admissible signed graphs $(G,\sigma)$ with two negative edges. We restrict our study to cubic graphs, because for each non-cubic signed graph $(G,\sigma)$ there is a set of cubic graphs obtained from $(G,\sigma)$ such that the flow number of $(G,\sigma)$ does not exceed the flow number of any of the cubic graphs. We prove that $F(G,\sigma) \leq 6$ if $(G,\sigma)$ contains a bridge, and $F(G,\sigma) \leq 7$ in general. We prove better bounds, if there is a cubic graph $(H,\sigma_H)$ obtained from $(G,\sigma)$ which satisfies some additional conditions. In particular, if $H$ is bipartite, then $F(G,\sigma) \leq 4$ and the bound is tight. If $H$ is $3$-edge-colorable or critical or if it has a sufficient cyclic edge-connectivity, then $F(G,\sigma) \leq 6$. Furthermore, if Tutte's $5$-Flow Conjecture is true, then $(G,\sigma)$ admits a nowhere-zero $6$-flow endowed with some strong properties.

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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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