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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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.