Decomposing highly connected graphs into paths of length five

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

doi:10.1016/j.dam.2016.08.001

Cited by 7 publications

(10 citation statements)

References 37 publications

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“…In this paper, we refine their result by pushing the required edge-connectivity down to 3k − 3, even for even numbers k, and strengthen it by giving a sharp bound on out-degrees. In particular, we strengthen the recent result in [15] toward this concept which improves the required edge-connectivity of several results in [2,5,6,23,28,29,31] toward decomposing a graph into isomorphic copies of a fixed tree.…”

supporting

confidence: 82%

Modulo orientations with bounded out-degrees

Hasanvand¹

2017

Preprint

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Let G be a graph, let k be a positive integer, and let p : V (G) → Z k be a mapping withThis result reduces the required edge-connectivity of several results toward decomposing a graph into isomorphic copies of a fixed tree. Next, we conclude that if G is a (3k − 3)-edge-connected bipartite graph with the bipartition (A, B), then it has a factor H such that for each vertex v,. Finally, we investigate decomposition of highly edge-connected graphs into factors with bounded degrees with many edge-disjoint spanning trees and deduce that every 4-edge-connected graph G has a spanning Eulerian subgraph whose degrees are close to dG(v)/2. As a consequence, every 4-edge-connected 10-regular graph has a spanning Eulerian subgraph whose degrees lie in the set {4, 6}.

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confidence: 82%

Modulo orientations with bounded out-degrees

Hasanvand¹

2017

Preprint

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“…For any tree T on m edges, there exists an integer k T such that every k T -edge-connected graph with size divisible by m has a Tdecomposition.When the conjecture was made, it was only known to hold in the trivial cases where T has less than 3 edges. Since then, Conjecture 1 has attracted growing attention, and it has now been verified for different families of trees such as stars [14], some bistars [1,16], and paths of a certain length [6,12,13,15]. Very recently, breakthrough results were obtained by Merker [10], who proved the conjecture for all trees of diameter at most 4, hence covering some of the results above, and by Botler, Mota, Oshiro, and Wakabayashi [5], who proved the conjecture for all paths.…”

mentioning

confidence: 99%

A proof of the Barát–Thomassen conjecture

Bensmail

Harutyunyan

Merker

et al. 2017

Journal of Combinatorial Theory, Series B

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The Barát-Thomassen conjecture asserts that for every tree T on m edges, there exists a constant k T such that every k T -edge-connected graph with size divisible by m can be edge-decomposed into copies of T . So far this conjecture has only been verified when T is a path or when T has diameter at most 4. Here we prove the full statement of the conjecture. condition is also sufficient in certain cases. By a result of Wilson [17] this holds when G is a sufficiently large complete graph, and there exist more general results showing that this is also true for graphs of large minimum degree. More precisely, for every tree T there exists a constant ε T > 0 such that every graph G of minimum degree (1 − ε T )|V (G)| admits a T -decomposition, provided its size is divisible by the size of T (see for example [2]).A different line of research was started by Barát and Thomassen [3], when they observed in 2006 that T -decompositions are intimately related to nowhere-zero flows. Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero 3-flow, but until recently it was not even known that any constant edge-connectivity suffices for this. Barát and Thomassen showed that if every 8-edge-connected graph of size divisible by 3 admits a K 1,3 -decomposition, then every 8-edge-connected graph admits a nowhere-zero 3-flow. Vice versa, they also showed that Tutte's 3-flow conjecture would imply that every 10-edgeconnected graph with size divisible by 3 admits a K 1,3 -decomposition. Motivated by this intrinsic connection, they conjectured the following.Conjecture 1 (Barát-Thomassen Conjecture, [3]). For any tree T on m edges, there exists an integer k T such that every k T -edge-connected graph with size divisible by m has a Tdecomposition.When the conjecture was made, it was only known to hold in the trivial cases where T has less than 3 edges. Since then, Conjecture 1 has attracted growing attention, and it has now been verified for different families of trees such as stars [14], some bistars [1,16], and paths of a certain length [6,12,13,15]. Very recently, breakthrough results were obtained by Merker [10], who proved the conjecture for all trees of diameter at most 4, hence covering some of the results above, and by Botler, Mota, Oshiro, and Wakabayashi [5], who proved the conjecture for all paths. The latter result was improved by Bensmail, Harutyunyan, Le, and Thomassé [4], who showed that, for path-decompositions, large minimum degree is a sufficient condition provided the graph is 24-edge-connected.The purpose of this paper is to verify Conjecture 1 for every tree T , hence settling the conjecture in the affirmative.Theorem 2. The Barát-Thomassen conjecture is true.This paper builds upon previous work on the Barát-Thomassen conjecture. It was shown by Thomassen in [16], and independently by Barát and Gerbner in [1], that it is sufficient to verify Conjecture 1 for bipartite graphs G.Theorem 3. [1,16] Let T be a tree on m edges. The following two statements are equivalent:(1) There exists a natural number k T...

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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: 94%

“…In [Botler et al 2016], we proved that Conjecture 3.1 holds for paths of length 5. This result was also obtained by Merker [Merker 2017], who, additionally, verified Conjecture 3.1 for trees with diameter at most 4.…”

Section: Decomposition Of Highly Edge-connected Graphs Into Paths Of mentioning

confidence: 95%

“…Later, we were able to explore better this technique to obtain path decomposition of two important family of graphs, namely, regular graphs and highly edge-connected graphs. Our proofs use a generalization of the technique we presented in [Botler et al 2016], which combines a method introduced by Thomassen [Thomassen 2008a] and a technique used by Lovász [Lovász 1968] for decomposition into cycles and paths.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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Decomposing highly connected graphs into paths of length five

Cited by 7 publications

References 37 publications

Modulo orientations with bounded out-degrees

Modulo orientations with bounded out-degrees

A proof of the Barát–Thomassen conjecture

Decomposition of Graphs into Paths

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