Simultaneous well-balanced orientations of graphs

Nash-Williamsʹ 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. ve that if G 1 , … , G k are pairwise edge-disjoint subgraphs of a graph G, then G has a well-balanced orientation G → such that the inherited orientations G → i of G i 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.