Cycle double covers and spanning minors II

Häggkvist, Roland; Markström, Klas

doi:10.1016/j.disc.2005.10.031

Cited by 5 publications

(6 citation statements)

References 8 publications

(21 reference statements)

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“…Unpublished work by Markström establishes the validity of the CDCC whenever G has a 2‐factor with exactly six components, and Häggkvist and Markström established that the CDCC is true whenever G has a 2‐factor F such that the contraction

G_{F}

has a circuit containing all odd vertices of

G_{F}

. Several articles, in particular Goddyn's PhD thesis and an article by Häggkvist and Markström tackle the CDCC via special types of spanning subgraphs called frames; see and the cited literature for details.…”

Section: Introduction and Preliminary Discussionmentioning

confidence: 99%

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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G_{F}

has a circuit containing all odd vertices of

G_{F}

Section: Introduction and Preliminary Discussionmentioning

confidence: 99%

Cycle Double Covers in Cubic Graphs having Special Structures

Fleischner

Häggkvist

2013

J. Graph Theory

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“…A cubic graph H is a spanning minor of a cubic graph G if some subdivision of H is a spanning subgraph of G. In [3], Goddyn showed that a cubic graph G has a circuit double cover if it contains the Petersen graph as a spanning minor. Goddyn's result is further improved by Häggkvist and Markström [6] who showed that a cubic graph G has a circuit double cover if it contains a 2connected simple cubic graph with no more than 10 vertices as a spanning minor.…”

Section: Introductionmentioning

confidence: 98%

“…According to their observations [5,6], Häggkvist and Markström proposed the following conjectures. Conjecture 1.3 (Häggkvist and Markström, [5]).…”

Section: Introductionmentioning

confidence: 99%

Cycle double covers and the semi-Kotzig frame

Zhang

2012

European Journal of Combinatorics

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Let H be a cubic graph admitting a 3-edge-coloring c : E(H) → Z3 such that the edges colored by 0 and µ ∈ {1, 2} induce a Hamilton circuit of H and the edges colored by 1 and 2 induce a 2-factor F . The graph H is semi-Kotzig if switching colors of edges in any even subgraph of F yields a new 3-edge-coloring of H having the same property as c. A spanning subgraph H of a cubic graph G is called a semi-Kotzig frame if the contracted graph G/H is even and every non-circuit component of H is a subdivision of a semi-Kotzig graph. In this paper, we show that a cubic graph G has a circuit double cover if it has a semi-Kotzig frame with at most one non-circuit component. Our result generalizes some results of Goddyn (1988), and Häggkvist and Markström [J. Combin. Theory Ser. B (2006)].

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“…Goddyn's result is further improved by Häggkvist and Markström [23] who showed that a cubic graph G has a circuit double cover if it contains a 2-connected simple cubic graph with no more than 10 vertices as a spanning minor. According to their observations [22,23], Häggkvist and Markström conjectured that every 3-connected cubic graph contains a Kotzig graph as a spanning minor.…”

Section: Introductionmentioning

confidence: 98%

“…Circuit Double Conjecture has been verified for K 5 -minor-free graphs, Petersenminor-free graphs [3] and graphs with specific structures such as Hamiltonian path [47], small oddness ( [27], [24], [26]) and spanning subgraphs ( [20], [22], [23], [51] etc).…”

Section: Introductionmentioning

confidence: 99%

Perfect Matching and Circuit Cover of Graphs

Ye¹

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Cycle double covers and spanning minors II

Cited by 5 publications

References 8 publications

Cycle Double Covers in Cubic Graphs having Special Structures

Cycle Double Covers in Cubic Graphs having Special Structures

Cycle double covers and the semi-Kotzig frame

Perfect Matching and Circuit Cover of Graphs

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