Optimal edge-colourings for a class of planar multigraphs

Let G be a multigraph containing no minor isomorphic to K-3,K-3 or K-5\e (w here K-5\e denotes K-5 without one of its edges). We show that the chromati c index of G is given by max{rho, [kappa]}, where rho is the maximum valenc y of G and kappa is defined as max{omega (E(S))/[\S \ /2] \S is an odd subset of vertices of G with \S \ g reater than or equal to 3} (omega (E(S)) being the number of edges in the subgraph induced by S). This result partially verifies a conjecture of Seymour [J. Combin. Theory (B) 3 1 (1981), pp. 82-94] and is actually a generalization of a result proven by Seymour [Combinatorica 10 (1990), pp. 379-392] for series-parallel graphs, It is also equivalent to the following statement: the matching polytope of a graph containing neither K-5\e nor K-3,K-3 as a minor has the integer de composition property.