  A binary Cayley graph is a Cayley graph based on a binary group. In 1982,
Payan proved that any non-bipartite binary Cayley graph must contain a
generalized Mycielski graph of an odd-cycle, implying that such a graph cannot
have chromatic number 3. We strengthen this result first by proving that any
non-bipartite binary Cayley graph must contain a projective cube as a subgraph.
We further conjecture that any homo- morphism of a non-bipartite binary Cayley
graph to a projective cube must be surjective and we prove some special case of
this conjecture.
