I am curios to know at least one example of the following graphs:
i) an infinite class of bipartite graphs that is randomly matchable;
ii) an infinite class of non-bipartite graphs that is randomly matchable;
ii) an example of a graph that has a 1-factor, but is not randomly matchable.
By condition, if every matching of the graph can be extended to a 1-factor then the graph is said to be a randomly matchable graph. Thus, in my opinion (which may be wrong), a Rado graph is an infinite class of bipartite graphs that is randomly matchable. Any help and guidance is welcome to lead me unravel the above graphs. Thanks in advance!