Graph orientations with edge-connection and parity constraints

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V, E) having a k-edge-connected T-odd orientation for every subset T subset of or equal to V with \E\ + \T\ even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge-connected graph with \V\ + \E\ even has a (k - 1)-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k. edge-disjoint paths from 8 to every node and k - 1 edge-disjoint paths from every node to s.