For a finite graph G, we study the maximum 2-edge colorable subgraph problem and a related ratio , where is the matching number of G, and is the size of the largest matching in any pair of disjoint matchings maximizing (equivalently, forming a maximum 2-edge colorable subgraph). Previously, it was shown that , and the class of graphs achieving was completely characterized. In this paper, we first show that graph decompositions into paths and even cycles provide a new way to study these parameters. We then use this technique to characterize the graphs achieving among all graphs that can be covered by a certain choice of a maximum matching and H, as above.
"Characterization of saturated graphs related to pairs of disjoint matchings." Illinois J. Math. 66 (1) 59 - 77, April 2022. https://doi.org/10.1215/00192082-9719963