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We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of error-correcting codes.… (More)

Let G and H be two simple, undirected g r aphs. An e m b edding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H, t o gether with a mapping which assigns to each edge uu v] of G a p ath between f(u) and f(v) in H. The grid M(r s) is the graph whose vertex set is the set of pairs on nonnegative integers,… (More)

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H. The dilation of the embedding is the maximum distance between f(u); f(v) taken over all edges (u; v) of G. We give a construction of embeddings of dilation 1 of complete binary trees into star… (More)