Edge-decomposition of graphs into copies of a tree with four edges

Abstract

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree T, there exists a natural number kT such that if G is a kT -edge-connected graph, and |E(T)| divides |E(G)|, then E(G) has a decomposition into copies of T. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013). Let Y be the unique tree with degree sequence (1, 1, 1, 2, 3). We prove that if G is a 191-edge-connected graph of size divisible by 4, then G has a Y -decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant. Keywords: graph theory; decomposition; tree; edge-connectivity.