i have searched google for your question and i think your question is equivalent to :

for which regular bipartite graphs do there exist quadrilateral embeddings?

so i find 3 similar papers for you :

1. http://www.ams.org/journals/tran/1970-151-02/S0002-9947-1970-0281653-3/

2. http://lib.gen.in/16bbcefe972e05a1cb422e8ff3b9bc73.pdf

3. www.sciencedirect.com/science/article/pii/0012365X89901702

is it helpful? thank you !