Decomposition of Graphs on Surfaces

Abstract

LetG=(V, E) be an Eulerian graph embedded on a triangulizable surfaceS. We show thatEcan be decomposed into closed curvesC1, …, Cksuch that mincr(G, D)=∑ki=1mincr(Ci, 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, …, Ckare closed curves inGsuch that each edge is traversed exactly once byC1, …, Ck. So each vertexvis traversed exactly 12deg(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, …, Ckbe closed curves onS. Then there exist circulationsf1, …, fk∈REhomotopic toC1, …, Ckrespectively such thatf1(e)+…+fk(e)⩽1 for each edgeeif and only if mincr(G, D)⩾∑ki=1mincr(Ci, D) for each closed curveDonS. Here acirculation homotopicto a closed curveC0is any convex combination of functions trC∈RE, whereCis a closed curve inGfreely homotopic toC0and where trC(e) is the number of timesCtraversese.