Thinking about this further last night, I can refine the requirement to finding: that union of cyclic alternating paths which puts each node in the longest possible cycle. This page you referenced uses the discovery of alternating paths at the core of its algorithm - I need to analyse it further to see if it can be extended to discern the longest cycles.

I was perhaps overly swayed by reading that the problem of "finding the longest cycle" is NP-hard - that is for a general graph, and it seems entirely possible that the special case of a bipartite graph introduces enough structure to make it rather easier.

Hugo