A class of matchings and a related lattice (Q1907106)

Let \(\eta' (G)\) denote the fractional chromatic index of a graph \(G\). Let \({\mathcal M} (G)\) be a family of matchings of \(G\) such that \(M \in {\mathcal M} (G)\) if and only if there exists a fractional \(\lceil \eta' (G) \rceil\)- edge colouring of \(G\) in which \(M\) is a colour class. We use Lovász's matching lattice theorem to obtain a description of the lattice generated by \({\mathcal M} (G)\). We show that, in a sense, the only complication arises when \(G\) has the Petersen graph minus a vertex as a minor.