  We study the ratio, in a finite graph, of the sizes of the largest matching
in any pair of disjoint matchings with the maximum total number of edges and
the largest possible matching. Previously, it was shown that this ratio is
between 4/5 and 1, and the class of graphs achieving 4/5 was completely
characterized. In this paper, we first show that graph decompositions into
paths and even cycles provide a new way to study this ratio. We then use this
technique to characterize the graphs achieving ratio 1 among all graphs that
can be covered by a certain choice of a maximum matching and maximum disjoint
matchings.
