A proof of the Barát–Thomassen conjecture

Abstract

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...

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

“…In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f (t)-edgeconnected graph with its number of edges divisible by t has a partition of its edges into copies of T . This conjecture was recently verified by the current authors and Merker [1].…”

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In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f (t)-edgeconnected graph with its number of edges divisible by t has a partition of its edges into copies of T . This conjecture was recently verified by the current authors and Merker [1].We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro and Wakabayashi [2].

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Random 4-regular graphs have 3-star decompositions asymptotically almost surely

“…Theorem 1.1 confirmed this when T is a star and indeed gives the best possible value of c T in that case. More recently, Bensmail, Harutyunyan, Le, Merker, and Thomassé [3] proved the conjecture for all trees T . However, determining what the best possible edge-connectivity is or, in the case of random regular graphs, what the best possible degree is, are still open problems.…”

“…Let B denote the number of cells that are centers in both σ 1 and σ 2 and have different special points. Note that A + B is maximized when all centers in σ 1 are centers in σ 2 as well; thus, A + B ≤ 2n 3 . We see that we may write E[Y 2 ] in terms of n, A, and B as follows.…”

“…√ 9a 2 + 48a. Let h(a) = (3a + 2b * (a)) 3 1 3 . Now one can check that the only zeros of h(a) with 0 ≤ a ≤ 2 3 are a = 0, a = 1 9 and a = 2 3 .…”