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.
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.