Graph homomorphism into an odd cycle

For graphs G and H, a map f : V (G) 7→ V (H) is a homomorphism if f preserves adjacency. Let Hom(G,H) denote the set of all homomorphisms from G into H. In this paper, we proved that for a simple graph G with n = |V (G)| and for k with n ≥ k ≥ 5, if the odd girth of G is at least 2k+1 and if the minimum degree δ(G) > 2n/(2k+3), then Hom(G,Z2k+1) 6= ∅, where Z2k+1 denotes the cycle of length 2k + 1. As a corollary, we settled affirmatively the following open problem posted by Alberson, Chan and Haas in 1993: If a graph G satisfies the conditions above, must the independence of G, which is the ratio of the independence number of G to the number of vertices of G, be at least k/(2k + 1)? ∗Part of this paper is done in this author’s Master thesis, and this author would like to give his special thanks to his Master superadvisor Professor Bolian Liu.