Abstract

LetG=(V, E) be an Eulerian graph embedded on a triangulizable surfaceS. We show thatEcan be decomposed into closed curvesC₁, …, C_ksuch that mincr(G, D)=∑^k_i=1 mincr(C_i, D) for each closed curveDonS. Here mincr(G, D) denotes the minimum number of intersections ofGandD′ (counting multiplicities), whereD′ ranges over all closed curvesD′ freely homotopic toDand not intersectingV. Moreover, mincr(C, D) denotes the minimum number of intersections ofC′ andD′ (counting multiplicities), whereC′ andD′ range over all closed curves freely homotopic toCandD, respectively.Decomposingthe edges means thatC₁, …, C_kare closed curves inGsuch that each edge is traversed exactly once byC₁, …, C_k. So each vertexvis traversed exactly  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 graphG=(V, E) on a compact surfaceS. LetC₁, …, C_kbe closed curves onS. Then there exist circulationsf₁, …, f_k ^Ehomotopic toC₁, …, C_krespectively such thatf₁(e)+…+f_k(e)1 for each edgeeif and only if mincr(G, D)∑^k_i=1mincr(C_i, D) for each closed curveDonS. Here acirculation homotopicto a closed curveC₀is any convex combination of functions tr_C ^E, whereCis a closed curve inGfreely homotopic toC₀and where tr_C(e) is the number of timesCtraversese.

^* Present address: Dr. Neher Laboratorium, P.O. Box 421, 2260 AK Leidschendam, The Netherlands.