The chromatic index of multigraphs of order at most 10

The maximum of the maximum degree and the 'odd set quotients' provides a well-known lower bound 4)(G) for the chromatic index of a multigraph G. Plantholt proved that if G is a multigraph of order at most 8, its chromatic index equals qS(G) and that if G is a multigraph of order 10, the chromatic index of G cannot exceed qS(G) + 1. We identify those multigraphs G of order 9 and 10 whose chromatic index equals ~b(G)+ 1, thus completing the determination of the chromatic index of all multigraphs of order at most 10.