Odd-angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G H → G T, then there is a graph homomorphism H → T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2 k + 1 )-angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)-cycle. We call G strongly (2k + 1)-angulated if every two vertices are connected by a sequence of (2k + 1)-cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)-angulated, H is any graph, S, T are graphs with odd girth at least 2k+ 1, and φ: G H → S T is a graph homomorphism, then either φ maps G {h} to S {t_h} for all h ∈ V(H) where t_h ∈ V(T) depends on h; or φ maps G {h} to {s_h} T for all h ∈ V(H) where s_h ∈ V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions.