Decomposing Graphs into a Spanning Tree, an Even Graph, and a Star Forest

Lyngsie, Kasper Szabo; Merker, Martin

doi:10.37236/7970

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…In the same year it was also proved to hold for claw-free (sub)cubic graphs by Aboomahigir, Ahanjideh, and Akbari [9] as well as by Hong et al [10]. More recently, Lyngsie and Merker [11] showed that weakening the matching requirement to allow for paths of length 2 suffices to make the conjecture true and Heinrich [12] verified it for 3-connected cubic graphs of tree-width 3. Earlier this year, Xie, Zhou, and Zhou [13] proved the conjecture for graphs with a two-factor consisting of three cycles.…”

Section: Introductionmentioning

confidence: 94%

Towards Obtaining a 3-Decomposition From a Perfect Matching

Bachtler

Krumke

2022

Electron. J. Combin.

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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: Introductionmentioning

confidence: 94%

Towards Obtaining a 3-Decomposition From a Perfect Matching

Bachtler

Krumke

2022

Electron. J. Combin.

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“…In the same year it was also proved to hold for claw-free (sub)cubic graphs by Aboomahigir, Ahanjideh, and Akbari [8]. 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%

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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