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Jul 22, 2010 at 21:37 comment added Joseph Malkevitch Gjergij: Yes, that is correct. While not noted above there is quite a large literature about the existence (or lack of existence) of cycles in graphs, in particular mod m. Perhaps things are easier in the case that interests you for planar graphs.
Jul 22, 2010 at 13:45 comment added Gjergji Zaimi This sounds interesting, however as I understand it these graphs do not provide a counterexample, that is they have some cycle of length 3m+1 for some m.
Jul 22, 2010 at 13:13 history answered Joseph Malkevitch CC BY-SA 2.5