Decomposing highly edge-connected graphs into paths of any given length

Abstract

Abstract. In 2006, Barát and Thomassen posed the following conjecture: for each tree T , there exists a natural number k T such that, if G is a k T -edge-connected graph and |E(G)| is divisible by |E(T )|, then G admits a decomposition into copies of T . This conjecture was verified for stars, some bistars, paths of length 3, 5, and 2 r for every positive integer r. We prove that this conjecture holds for paths of any fixed length.

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

“…In particular, our proof of Theorem 1.3 yields a third proof of the path case of Conjecture 1.1. It is also important mentioning that this proof is, in terms of approach, quite different from the one from [2].…”

“…Thomassen then proved, in [13], the conjecture for arbitrarily long paths of the form P 2 k . Later on, Botler, Mota, Oshiro and Wakabayashi proved the conjecture for P 5 [3] before generalizing their arguments and settling the conjecture for all paths [2]. Conjecture 1.1 being now solved, many related lines of research sound quite appealing.…”

“…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]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f (t) has an edge-partition into paths of length t whenever t divides the number of edges.…”

Decomposing 8-regular graphs into paths of length 4

“…) and by Hang(v, B) the number of edges leaving v in O i.e., Hang(v, B) = d + O (v) . We say that an edge that leaves v in O is a hanging edge at v (this definition coincides with the definition of pre-hanging edge in [1]). We say that a tracking decomposition…”

Hamilton path decompositions of complete multipartite graphs

“…Since Thomassen's paper [16], which shows that 171-edge-connected graphs of size divisible by 3 can be decomposed into paths of length 3, there has been considerable interest in path decompositions of highly connected graphs (also see [17]). In [4] it is shown that for any fixed length k, there is a constant C(k) such that any C(k)-edge-connected graph G has a decomposition into paths of length k if and only if k divides the size of G.…”