Simultaneous well-balanced orientations of graphs

Nash-Williams' well-balanced orientation theorem [C.St.J.A. Nash-Williams, On orientations, connectivity and odd-vertex-pairings in finite graphs, Canad. J. Math. 12 (1960) 555–567] is extended for orienting several graphs simultaneously.We prove that if G1,…,Gk are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation such that the inherited orientations of Gi are well-balanced for all 1⩽i⩽k. We also have new results about simultaneous well-balanced orientations of non-disjoint subgraphs of an Eulerian graph as well as those of different contractions of a graph.