Unique graph homomorphisms onto odd cycles, II

AbstractA natural generalization of graph colorings is graph homomorphisms. Let G and H be simple graphs. A map θ: V(G) → V(H) is called a homomorphism if θ preserves adjacency. The set of all homomorphism from G to H is denoted by Hom(G, H). A graph G is uniquely H-colorable if Hom(G, H) ≠ Π, and if for θ1, θ2 ∈ Hom(G, H), there is an automorphism π of H such that πθ1 = θ2. In this paper, we investigate some necessary necessary conditions of unique C2k+1-colorings and prove a best possible sufficient condition involving δ(G) for G to be uniquely C2k+1-colorable under some necessary conditions. This generalizes a result of Bollobás on unique C3-colorings [J. Combin. Theory Ser. B 25 (1978), 55–61]. We also find best possible conditions on the connectedness of the subgraphs of G induced by the preimages of θ, for any θ ∈ Hom(G, C2k+1)