Between 2- and 3-Colorability

We consider the question of the existence of homomorphisms between Gn,p and odd cycles when p = c/n, 1 < c ≤ 4. We show that for any positive integer l, there exists ε = ε(l) such that if c = 1 + ε then w.h.p. Gn,p has a homomorphism from Gn,p to C2l+1 so long as its odd-girth is at least 2l+1. On the other hand, we show that if c = 4 then w.h.p. there is no homomorphism from Gn,p to C5. Note that in our range of interest, χ(Gn,p) = 3 w.h.p., implying that there is a homomorphism from Gn,p to C3. These results imply the existence of random graphs with circular chromatic numbers χc satisfying 2 < χc(G) < 2 + δ for arbitrarily small δ, and also that 2.5 ≤ χc(Gn, 4 n ) < 3 w.h.p.