Generation and properties of snarks

Brinkmann, Gunnar; Goedgebeur, Jan; Hägglund, Jonas; Markström, Klas

doi:10.1016/j.jctb.2013.05.001

Cited by 98 publications

(182 citation statements)

References 42 publications

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“…Then the graph

\overset{H}{true}

obtained from

G_{0}, G_{1}, G_{2}

by the windmill construction is a snark such that:

scc (\overset{H}{true}) = \frac{4}{3} | E (\overset{˚}{H}) | + 1

For every cycle cover

scriptC

\overset{H}{true}

of minimum length, the vertex

t_{C}

cannot be one of the vertices

c_{i}

(see Fig. ). Proof The first part of the lemma was proved in . Assume now that the second part fails for some cycle cover

scriptC

of length

\frac{4}{3} | E (\overset{˚}{H}) | + 1

, and let P be the 1‐parity subgraph associated to

scriptC

.…”

Section: Shortest Cycle Covermentioning

confidence: 99%

“…Recall that beside the Petersen graph, only one snark G with

scc (G) > \frac{4}{3} | E (G) |

was known (and the proof of it was computer‐assisted). Theorem has the following immediate corollary: Corollary The family

scriptF

is an infinite family of snarks such that for any

G \in F

scc (G) > \frac{4}{3} | E (G) |

.…”

Section: Shortest Cycle Covermentioning

confidence: 99%

“…On the other hand, recall that Brinkmann et al. conjectured that every snark G has a cycle cover of size at most .…”

Section: Open Problemsmentioning

confidence: 99%

“…Brinkmann et al. proved using a computer program that the only snarks G on m edges and at most 36 vertices having no cycle cover of length are the Petersen graph and the 34‐vertex graph

\overset{H}{true}

mentioned above. Moreover, these two graphs have a cycle cover of length .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Mazzuoccolo

2013

J. Graph Theory

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The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known.It turns out that the family F also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with m edges has length at least 4 3 m, and we show that this inequality is strict for graphs of F. We also construct the first known snark with no cycle cover of length less than 4 3 m + 2.

show abstract

“…Then the graph

\overset{H}{true}

obtained from

G_{0}, G_{1}, G_{2}

by the windmill construction is a snark such that:

scc (\overset{H}{true}) = \frac{4}{3} | E (\overset{˚}{H}) | + 1

For every cycle cover

scriptC

\overset{H}{true}

of minimum length, the vertex

t_{C}

cannot be one of the vertices

c_{i}

(see Fig. ). Proof The first part of the lemma was proved in . Assume now that the second part fails for some cycle cover

scriptC

of length

\frac{4}{3} | E (\overset{˚}{H}) | + 1

, and let P be the 1‐parity subgraph associated to

scriptC

.…”

Section: Shortest Cycle Covermentioning

confidence: 99%

“…Recall that beside the Petersen graph, only one snark G with

scc (G) > \frac{4}{3} | E (G) |

was known (and the proof of it was computer‐assisted). Theorem has the following immediate corollary: Corollary The family

scriptF

is an infinite family of snarks such that for any

G \in F

scc (G) > \frac{4}{3} | E (G) |

.…”

Section: Shortest Cycle Covermentioning

confidence: 99%

“…On the other hand, recall that Brinkmann et al. conjectured that every snark G has a cycle cover of size at most .…”

Section: Open Problemsmentioning

confidence: 99%

“…Brinkmann et al. proved using a computer program that the only snarks G on m edges and at most 36 vertices having no cycle cover of length are the Petersen graph and the 34‐vertex graph

\overset{H}{true}

mentioned above. Moreover, these two graphs have a cycle cover of length .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Mazzuoccolo

2013

J. Graph Theory

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show abstract

“…Provided one had a means of enumeration, an extension of this program to other classes would be straightforward. Exhaustive generation algorithms are known for many specialized classes such as bipartite graphs, digraphs, multigraphs, regular graphs, cubic graphs, snarks, trees and maximal triangle-free graphs[14, 15, 16, 17, 18, 19]. …”

Section: Introductionmentioning

confidence: 99%

Integer sequence discovery from small graphs

Hoppe

Petrone

2016

Discrete Applied Mathematics

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We have exhaustively enumerated all simple, connected graphs of a finite order and have computed a selection of invariants over this set. Integer sequences were constructed from these invariants and checked against the Online Encyclopedia of Integer Sequences (OEIS). 141 new sequences were added and six sequences were extended. From the graph database, we were able to programmatically suggest relationships among the invariants. It will be shown that we can readily visualize any sequence of graphs with a given criteria. The code has been released as an open-source framework for further analysis and the database was constructed to be extensible to invariants not considered in this work.

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Cycle Double Covers in Cubic Graphs having Special Structures

Fleischner

Häggkvist

2013

J. Graph Theory

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In the first part of this article, we employ Thomason's Lollipop Lemma to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2‐factor F having two components admit CDCs containing any of the components in the 2‐factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2‐factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k‐circuit C together with an induced 4k‐circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.

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Generation and properties of snarks

Cited by 98 publications

References 42 publications

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

Integer sequence discovery from small graphs

Cycle Double Covers in Cubic Graphs having Special Structures

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