Decomposition of Graphs on Surfaces

Let G=(V, E) be an Eulerian graph embedded on a triangulizable surface S. We show that E can be decomposed into closed curves C1 , ..., Ck such that mincr(G, D)= i=1 mincr(Ci , D) for each closed curve D on S. Here mincr(G, D) denotes the minimum number of intersections of G and D$ (counting multiplicities), where D$ ranges over all closed curves D$ freely homotopic to D and not intersecting V. Moreover, mincr(C, D) denotes the minimum number of intersections of C$ and D$ (counting multiplicities), where C$ and D$ range over all closed curves freely homotopic to C and D, respectively. Decomposing the edges means that C1 , ..., Ck are closed curves in G such that each edge is traversed exactly once by C1 , ..., Ck . So each vertex v is traversed exactly 2 deg (v) times, where deg (v) is the degree of v. This result was shown by Lins for the projective plane and by Schrijver for compact orientable surfaces. The present paper gives a shorter proof than the one given for compact orientable surfaces. We derive the following fractional packing result for closed curves of given homotopies in a graph G=(V, E) on a compact surface S. Let C1 , ..., Ck be closed curves on S. Then there exist circulations f1 , ..., fk # R homotopic to C1 , ..., Ck respectively such that f1(e)+ } } } + fk(e) 1 for each edge e if and only if mincr(G, D) i=1 mincr(Ci , D) for each closed curve D on S. Here a circulation homotopic to a closed curve C0 is any convex combination of functions trC # R , where C is a closed curve in G freely homotopic to C0 and where trC(e) is the number of times C traverses e. 1997 Academic Press article no. TB971747