Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

Let G be a 2k‐edge‐connected graph with k≥0 and let Lfalse(vfalse)⊆{k,…,dGfalse(vfalse)} for every v∈V(G). A spanning subgraph F of G is called an L‐factor, if dFfalse(vfalse)∈Lfalse(vfalse) for every v∈V(G). In this article, we show that if false|L(v)false|≥false⌈dG(v)2false⌉+1 for every v∈V(G), then G has a k‐edge‐connected L‐factor. We also show that if k≥1 and Lfalse(vfalse)={⌊dG(v)2⌋,…,false⌈dG(v)2false⌉+k} for every v∈V(G), then G has a k‐edge‐connected L‐factor.

show abstract

“…Theorem directly gives the following corollary. This statement was already shown in , and implicitly in for

k = 2, 6

. Corollary Every 2 k ‐edge‐connected graph G has a spanning k ‐tree‐connected subgraph H such that for every

v \in V (H)

d_{H} false(v false) \leq false⌈ \frac{d_{G} (v)}{2} false⌉ + k

.…”

Section: Highly Edge‐connected L‐factors With Large Listsmentioning

confidence: 64%

Highly edge‐connected factors using given lists on degrees

Akbari

Hasanvand

Ozeki

2018

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…However, if the trees in N I (v|t) overlap too much, then we might not be able to make any switch that improves the T -pseudo-decomposition. To avoid this, we need to find a large set of isomorphic copies in N I (v|t) that pairwise intersect only in v. To measure how much the pseudo-trees in a T -pseudo-decomposition overlap, we use the following concept that was introduced in [4].…”

Section: Definitions and Sketch Of Proofmentioning

confidence: 99%

“…Now we apply Proposition 16 to show that with positive probability none of the events A v,t ′ occur. Set x = ε 4 8 . It is sufficient to show that…”

Section: Repairing Non-isomorphic Copiesmentioning

confidence: 99%

See 1 more Smart Citation

A proof of the Barát–Thomassen conjecture

Bensmail

Harutyunyan

Merker

et al. 2017

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…A series of results showing that Conjecture 1 holds for specific trees followed [1,14,13,8,10,11,5,6,12,4]. Recently, the conjecture was proven by Bensmail, Harutyunyan, Le, Merker and the second author [3].…”

Section: Introductionmentioning

confidence: 99%

Edge‐decomposing graphs into coprime forests

Klimošová

Thomassé

2020

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

The Barát‐Thomassen conjecture, recently proved in Bensmail et al. (2017), asserts that for every tree T, there is a constant c T such that every c T‐edge‐connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T‐decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in Bensmail et al. (2019) that when T is a path with k edges, there is a constant d k such that every 24‐edge‐connected graph G with size divisible by k and minimum degree d k has a T‐decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F‐decomposition provided that the size of F divides the size of G (no connectivity is required). A natural conjecture asked in Bensmail et al. (2019) asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T‐decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3.

show abstract

Edge-Partitioning a Graph into Paths: Beyond the Barát-Thomassen Conjecture

Cited by 13 publications

References 14 publications

Highly edge‐connected factors using given lists on degrees

Highly edge‐connected factors using given lists on degrees

A proof of the Barát–Thomassen conjecture

Edge‐decomposing graphs into coprime forests

Contact Info

Product

Resources

About