Decomposing regular graphs with prescribed girth into paths of given length

Botler, Fábio; Mota, Guilherme Oliveira; Oshiro, M. T. I.; Wakabayashi, Yoshiko

doi:10.1016/j.ejc.2017.06.011

Cited by 10 publications

(5 citation statements)

References 12 publications

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“…We conclude mentioning that the results obtained in [Botler 2016] have been published in the Journal of Combinatorial Theory, Series B [Botler et al 2017a] and Discrete Mathematics [Botler et al 2015c]; and have been accepted to the European Journal of Combinatorics [Botler et al 2017b] and Discrete Applied Mathematics [Botler et al 2016], the first one being one the most prestigious journals in combinatorics. We have also presented these results in many international conferences, among which we mention ICGT 2014, LAGOS 2015, EuroComb 2015 [Botler et al 2015b, Botler et al 2015a.…”

Section: Discussionmentioning

confidence: 93%

“…In [Botler et al 2017b], we also prove the following result, which is a weakening of Conjecture 2.5: for each positive integers and g such that is odd and g ≥ 3, there is an m 0 = m 0 ( , g) such that, if G is an m -regular graph with m ≥ m 0 , girth at least g, and containing an m-factor, then G admits a balanced P -decomposition. We also give a bound for m 0 .…”

Section: Decomposition Of Regular Graphs Into Paths Of Fixed Lengthmentioning

confidence: 88%

“…In [Botler and Talon 2017] it is proved that Conjecture 2.2 holds for = 4 (see Theorem 2.4). In [Botler et al 2017b] we prove a weakening of Conjecture 2.2, which states that, for each positive integers and g such that g ≥ 3, there is an m 0 = m 0 ( , g) such that, if G is a 2m -regular graph with m ≥ m 0 and girth at least g, then G admits a balanced P -decomposition. The next theorem gives a bound for m 0 .…”

Section: Decomposition Of Regular Graphs Into Paths Of Fixed Lengthmentioning

confidence: 99%

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Decomposition of Graphs into Paths

Botler¹,

Wakabayashi²

2017

Anais Do XXX Concurso De Teses E Dissertações (CTD 2017)

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We study the Decomposition Conjecture posed by Barát and Thomassen (2006), which states that, for each tree T , there exists a natural number kT such that, if G is a kT -edge-connected graph and jE(T )j divides jE(G)j, then G admits a partition of its edge set into copies of T . In a series of papers, Thomassen has verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. In this paper we prove this conjecture for paths of any given length. Our technique is then used to prove weakenings of a conjecture of Kouider and Lonc (1999), and a conjecture of Favaron, Genest and Kouider (2010), both for path decomposition of regular graphs.

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Section: Discussionmentioning

confidence: 93%

Section: Decomposition Of Regular Graphs Into Paths Of Fixed Lengthmentioning

confidence: 88%

Section: Decomposition Of Regular Graphs Into Paths Of Fixed Lengthmentioning

confidence: 99%

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Decomposition of Graphs into Paths

Botler¹,

Wakabayashi²

2017

Anais Do XXX Concurso De Teses E Dissertações (CTD 2017)

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“…In 2015, Botler, Mota, and Wakabayashi [3] verified Conjecture 1 for triangle-free 5-regular, and, more recently, Botler, Mota, Oshiro, and Wakabayashi [2] generalized this result for (2k + 1)-regular graphs with girth at least 2k.…”

Section: Introductionmentioning

confidence: 93%

Decomposition of (2k+1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k+1

Botler¹,

Hoffmann²

2020

Anais Do Encontro De Teoria Da Computação (ETC 2020)

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A Pl-decomposition of a graph G is a set of paths with l edges in G that cover the edge set of G. Favaron, Genest, and Kouider (2010) conjectured that every (2k+1)-regular graph that contains a perfect matching admits a P2k+1-decomposition. They also verified this conjecture for 5-regular graphs without cycles of length 4. In 2015, Botler, Mota, and Wakabayashi extended this result to 5-regular graphs without triangles. In this paper, we verify this conjecture for (2k+1)-regular graphs that contain the k-th power of a spanning cycle; and for 5-regular graphs that contain certain spanning 4-regular Cayley graphs.

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“…One of the authors [2] proved the following weakening of Conjecture 1.2: for every positive integers ℓ and g such that g ≥ 3, there exists an integer m 0 = m 0 (ℓ, g) such that, if G is a 2mℓ-regular graph with m ≥ m 0 , then G admits a P ℓ -decomposition D such that every vertex of G is the end-vertex of exactly 2m paths of D. In this paper we prove Conjecture 1.2 in the case ℓ = 4.…”

Section: Introductionmentioning

confidence: 99%

Decomposing 8-regular graphs into paths of length 4

Botler

Talon

2017

Discrete Mathematics

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A T -decomposition of a graph G is a set of edge-disjoint copies of T in G that cover the edge set of G. Graham and Häggkvist (1989) conjectured that any 2ℓ-regular graph G admits a T -decomposition if T is a tree with ℓ edges. Kouider and Lonc (1999) conjectured that, in the special case where T is the path with ℓ edges, G admits a T -decomposition D where every vertex of G is the end-vertex of exactly two paths of D, and proved that this statement holds when G has girth at least (ℓ + 3)/2. In this paper we verify Kouider and Lonc's Conjecture for paths of length 4.

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Decomposing regular graphs with prescribed girth into paths of given length

Cited by 10 publications

References 12 publications

Decomposition of Graphs into Paths

Decomposition of Graphs into Paths

Decomposition of (2k+1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k+1

Decomposing 8-regular graphs into paths of length 4

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