A Note on 5-Cycle Double Covers

Hoffmann-Ostenhof, Arthur

doi:10.1007/s00373-012-1169-8

Cited by 6 publications

(6 citation statements)

References 2 publications

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“…In a common refinement of the strong CDCC and the 5-CDCC one may ask whether any cycle is part of some 5-CDC. In [29] Hoffmann-Ostenhof gives the following characterization of 2-regular subgraphs which can be part of a 5-CDC: Theorem 6.4 (Hoffmann-Ostenhof [29]). Let G be a cubic graph and D be a 2-regular subgraph of G. Then there is a 5-CDC which contains D if and only if G contains two 2-regular subgraphs D 1 and D 2 such that D ⊂ D 1 , M = E(D 1 ) ∩ E(D 2 ) is a matching and G − M has a nowhere-zero 4-flow.…”

Section: Oriented Cycle Double Covers and K-cdcsmentioning

confidence: 99%

“…In the same paper he conjectured that not only can every cycle be included in some CDC but it can in fact always be part of some 5-CDC. Conjecture 6.5 (Hoffmann-Ostenhof [29]). Every cycle C in a bridgeless cubic graph G is contained in some 5-CDC of G.…”

Section: Oriented Cycle Double Covers and K-cdcsmentioning

confidence: 99%

“…Figure 4. A snark where the 2-factor given by the cycles [1,4,9,5,2] and [3,8,14,22,23,15,24,30,25,17,18,12,6,11,13,20,21,27,19,28,29,26,16,10,7] cannot be part of any 5-CDC.…”

Section: Oriented Cycle Double Covers and K-cdcsmentioning

confidence: 99%

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Generation and properties of snarks

Brinkmann

Goedgebeur

Hägglund

et al. 2013

Journal of Combinatorial Theory, Series B

170

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For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for snarks, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured.In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on n ≤ 36 vertices. Previously lists up to n = 28 vertices have been published.In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.

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Section: Oriented Cycle Double Covers and K-cdcsmentioning

confidence: 99%

Section: Oriented Cycle Double Covers and K-cdcsmentioning

confidence: 99%

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Generation and properties of snarks

Brinkmann

Goedgebeur

Hägglund

et al. 2013

Journal of Combinatorial Theory, Series B

170

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“…G in Corollary 4.3 has some CDC even if C is separating, see [10]. The above results offer some insight which cycles are part of a 5-CDC, see the Strong 5-CDCC in [14].…”

Section: Proofmentioning

confidence: 85%

Cycle double covers and non-separating cycles

Hoffmann-Ostenhof

Zhang

2019

European Journal of Combinatorics

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Which 2-regular subgraph R of a cubic graph G can be extended to a cycle double cover of G? We provide a condition which ensures that every R satisfying this condition is part of a cycle double cover of G. As one consequence, we prove that every 2-connected cubic graph which has a decomposition into a spanning tree and a 2-regular subgraph C consisting of k circuits with k ≤ 3, has a cycle double cover containing C.

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“…As an approach toward proving the CDCC Goddyn suggested that every circuit in a bridgeless cubic graph could be used to start a CDC (this is commonly referred to as the strong cycle double cover conjecture, SCDCC), and proved this for the Isaacs snarks. In view of this and Celmins' 5CDCC, Hoffmann‐Ostenhof conjectured in his PhD thesis at the University of Vienna and in that every circuit in a bridgeless cubic graph can be used to start a 5CDC. This conjecture (S5CDCC) has been proven by a computer search for graphs with at most 34 vertices and for all cyclically 5‐edge connected snarks on 36 vertices, as reported in and .…”

Section: Introduction and Preliminary Discussionmentioning

confidence: 95%

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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A Note on 5-Cycle Double Covers

Cited by 6 publications

References 2 publications

Generation and properties of snarks

Generation and properties of snarks

Cycle double covers and non-separating cycles

Cycle Double Covers in Cubic Graphs having Special Structures

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