A Class of Matchings and a Related Lattice

Letη′(G) denote the fractional chromatic index of a graphG. Let M(G) be a family of matchings ofGsuch thatM∈M(G) if and only if there exists a fractionalη′(G)-edge colouring ofGin whichMis a colour class. We use Lovászʹs matching lattice theorem to obtain a description of the lattice generated by M(G). We shall show that, in a sense, the only complication arises whenGhas the Petersen graph minus a vertex as a minor.