Decomposition of cubic graphs with a 2-factor consisting of three cycles

Xie, Mengmeng; Zhou, Chuixiang; Zhou, Shun

doi:10.1016/j.disc.2020.111839

Cited by 4 publications

(11 citation statements)

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“…Here we remark that this lemma was also obtained by Xie, Zhou, and Zhou and can be found in [11,Lemma 2.3]. Their formulation basically describes the two cases in the proof, as they claim to either get a decomposition containing a cycle with two chords or a Hamiltonian path.…”

Section: Finding Decompositions In Cyclesmentioning

confidence: 55%

“…In Section 3 we present types of decompositions we can find in cycles. This has striking similarities to the techniques used in [11], which we describe in more detail when they occur. Using these we construct a 3-decomposition of a 3-connected star-like graph in Section 4.…”

Section: Introductionmentioning

confidence: 70%

“…More recently, Lyngsie and Merker [9] showed that weakening the matching requirement to allow for paths of length 2 suffices to make the conjecture true and Heinrich [10] proved the conjecture for 3-connected cubic graphs of tree-width 3. Earlier this year, Xie, Zhou, and Zhou [11] proved its validity when the graph has a two-factor consisting of three cycles.…”

Section: Introductionmentioning

confidence: 99%

“…Since many conjectures in graph theory, a prominent example being the cycle double cover conjecture [13], consider or can be reduced to bridgeless cubic graphs, obtaining structural information about these is of interest. Also, using this definition, the main theorem in [11] now reads that the conjecture is satisfied for any connected cubic graph with a perfect matching such that its contraction graph has order 3. This extends the previous proofs for Hamiltonian and traceable graphs, which handle contraction graphs of orders 1 and 2.…”

Section: Introductionmentioning

confidence: 99%

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Towards obtaining a 3-Decomposition from a perfect Matching

Bachtler¹,

Krumke²

2020

Preprint

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A decomposition of a graph is a set of subgraphs whose edges partition those of G. The 3-decomposition conjecture posed by Hoffmann-Ostenhof in 2011 states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph, and a matching. It has been settled for special classes of graphs, one of the first results being for Hamiltonian graphs. In the past two years several new results have been obtained, adding the classes of plane, claw-free, and 3-connected tree-width 3 graphs to the list.In this paper, we regard a natural extension of Hamiltonian graphs: removing a Hamiltonian cycle from a cubic graph leaves a perfect matching. Conversely, removing a perfect matching M from a cubic graph G leaves a disjoint union of cycles. Contracting these cycles yields a new graph G M . The graph G is star-like if G M is a star for some perfect matching M , making Hamiltonian graphs star-like. We extend the technique used to prove that Hamiltonian graphs satisfy the 3-decomposition conjecture to show that 3-connected star-like graphs satisfy it as well.

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Section: Finding Decompositions In Cyclesmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Towards obtaining a 3-Decomposition from a perfect Matching

Bachtler¹,

Krumke²

2020

Preprint

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“…If a cubic graph G decomposes into a spanning tree T , a 2-regular graph C, and a matching M , then (T, C, M ) is a 3-decomposition of G. Until the present day, the 3-decomposition conjecture remains wide open. The conjecture has been proved for a variety of subclasses of cubic graphs, for example planar cubic graphs [HOKO18], 3-connected cubic graphs of tree-width 3 [Hei19], traceable cubic graphs [AAHM16] (a graph is traceable if it admits a Hamiltonian path), and generalised Hamiltonian cubic graphs, see [BK20] and [XZZ20]. All of these results exploit one of the following two approaches:…”

Section: Introductionmentioning

confidence: 99%

Reductions for the 3-Decomposition Conjecture

Bachtler¹,

Heinrich²

2021

Preprint

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The 3-decomposition conjecture is wide open. It asserts that every cubic graph can be decomposed into a spanning tree, a disjoint union of cycles, and a matching. We show that every such decomposition is derived from a homeomorphically irreducible spanning tree (HIST), which is surprising since a graph with a HIST trivially satisfies the conjecture.We also prove that the following graphs are reducible configurations with respect to the 3-decomposition conjecture: the Petersen graph with one vertex removed, the claw-square, the twin-house, and the domino. As an application, we show that all graphs of path-width at most 4 satisfy the 3-decomposition conjecture and that a 3-connected minimum counterexample to the conjecture is triangle-free, all cycles of length at most 6 are induced, and every edge is in the centre of an induced P6. Finally, employing a computer, we prove that all graphs of order at most 20 satisfy the 3-decomposition conjecture.

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