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 2 k  + 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 (2 k  + 1)‐cycle. We call G strongly (2 k  + 1)‐angulated if every two vertices are connected by a sequence of (2 k  + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2 k  + 1)‐angulated, H is any graph, S , T are graphs with odd girth at least 2 k  + 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. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008