Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

Merker, Martin

doi:10.1016/j.jctb.2016.05.005

Cited by 13 publications

(25 citation statements)

References 27 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.…”

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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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“…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. Finally, in [Botler et al 2017a], we proved Conjecture 3.1 for paths of any given length.…”

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

confidence: 79%

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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“…Assuming G is (improperly) edge-coloured, we denote by d i (v) the degree of vertex v in colour i. For t ∈ V (T ), let S(t) denote the set of edges incident with t. It was shown by Merker [10], using modulo-k orientations, that highly edge-connected graphs admit T -equitable edge-colourings. Theorem 6 (Theorem 3.4 in [10]).…”

Section: Introductionmentioning

confidence: 99%

“…For t ∈ V (T ), let S(t) denote the set of edges incident with t. It was shown by Merker [10], using modulo-k orientations, that highly edge-connected graphs admit T -equitable edge-colourings. Theorem 6 (Theorem 3.4 in [10]). For all natural numbers m and L there exists a natural number f (m, L) such that the following holds: If G is an f (m, L)-edge-connected bipartite graph with bipartite classes A and B where all vertices in A have degree divisible by m, and T is a tree on m edges with bipartite classes T A and T B where T B contains a leaf, then G admits a T -equitable colouring where the minimum degree in each colour is at least L.…”

Section: Introductionmentioning

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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Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree

Cited by 13 publications

References 27 publications

Modulo orientations with bounded out-degrees

Modulo orientations with bounded out-degrees

Decomposition of Graphs into Paths

A proof of the Barát–Thomassen conjecture

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