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Recall that a closed walk (in a undirected graph) is a cycle if its vertices are pairwise distinct.

Does there exist random constructions of bipartite graphs without cycles with high probability?

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You might be looking for LERW and UST. –  Did Jun 4 '12 at 10:29

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up vote 2 down vote accepted

If a graph has no cycles then it is clearly bipartite. Moreover a graph without cycles is a forest. So what you really want is generate trees/forests?

If you're looking to generate random labeled forests/trees then this can be done efficiently using Prüfer sequences. Every such sequence chosen at random gives you a specific labeled tree.

Random non labeled forest are a bit harder to generate.

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