Claw-decompositions and tutte-orientations

We conjecture that, for each tree T , there exists a natural number kT such that the following holds: If G is a kT -edge-connected graph such that |E(T )| divides |E(G)|, then the edges of G can be divided into parts, each of which is isomorphic to T . We prove that for T = K1,3 (the claw), this holds if and only if there exists a (smallest) natural number kt such that every kt-edge-connected graph has an orientation for which the indegree of each vertex equals its outdegree modulo 3. Tutte’s 3-flow conjecture says that kt = 4. We prove the weaker statement that every 4dlog ne-edge-connected graph with n vertices has an edge-decomposition into claws provided its number of edges is divisible by 3. We also prove that every triangulation of a surface has an edge-decomposition into claws.