scholarly journals Treelike Snarks

10.37236/6008 ◽  
2016 ◽  
Vol 23 (3) ◽  
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.

Perfect Matching Index versus Circular Flow Number of a Cubic Graph

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

scholarly journals Circular flow number of generalized Blanuša snarks

2013 ◽  
Vol 313 (8) ◽  
pp. 975-981 ◽  
Author(s):  
Robert Lukot’ka
Keyword(s):  
Flow Number ◽  
Circular Flow

scholarly journals An infinite family of graphs with a facile count of perfect matchings

2014 ◽  
Vol 166 ◽  
pp. 210-214 ◽  
Author(s):  
Vladimir R. Rosenfeld ◽  
Douglas J. Klein
Keyword(s):  
Infinite Family ◽  
Perfect Matchings

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

2016 ◽  
Vol 7 (2–3) ◽  
pp. 453-479
Author(s):  
Louis Esperet ◽  
Giuseppe Mazzuoccolo ◽  
Michael Tarsi
Keyword(s):  
Recognition Problem ◽  
Flow Number ◽  
Circular Flow

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

COMBINATORICA ◽  
2002 ◽  
Vol 22 (3) ◽  
pp. 455-459 ◽  
Author(s):  
Anna Galluccio ◽  
Luis A. Goddyn
Keyword(s):  
Connected Graph ◽  
Flow Number ◽  
Circular Flow

scholarly journals Phylogenetic Trees, Augmented Perfect Matchings, and a Thron-type Continued Fraction (T-fraction) for the Ward Polynomials

10.37236/9571 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrew Elvey Price ◽  
Alan D. Sokal
Keyword(s):  
Generating Function ◽  
Phylogenetic Trees ◽  
Continued Fraction ◽  
Infinite Family ◽  
Perfect Matchings ◽  
Dyck Paths ◽  
Schröder Paths

We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.

scholarly journals Computational results and new bounds for the circular flow number of snarks

2020 ◽  
Vol 343 (10) ◽  
pp. 112026
Author(s):  
Jan Goedgebeur ◽  
Davide Mattiolo ◽  
Giuseppe Mazzuoccolo
Keyword(s):  
Computational Results ◽  
Flow Number ◽  
Circular Flow

scholarly journals Determining the Circular Flow Number of a Cubic Graph

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.

scholarly journals On Snarks that are far from being 3-Edge Colorable

10.37236/3430 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jonas Hägglund
Keyword(s):  
Double Cover ◽  
Interesting Property ◽  
Perfect Matchings ◽  
Petersen Graph ◽  
Cubic Graph

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

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