For a finite graph
, we
study the maximum
-edge
colorable subgraph problem and a related ratio
, where
is the matching
number of
,
and
is the size of the largest matching in any pair
of disjoint matchings
maximizing
(equivalently,
forming a maximum
-edge
colorable subgraph). Previously, it was shown that
, and the class of
graphs achieving
was completely characterized. We show here that any rational number between
and
can be
achieved by a connected graph. Furthermore, we prove that every graph with ratio less
than
must admit special subgraphs.
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