a b s t r a c tA function between graphs is k-to-1 if each point in the co-domain has precisely k preimages in the domain. Given two graphs, G and H, and an integer k ≥ 1, and considering G and H as subsets of R 3 , there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous k-to-1 functions between two intervals, each of which is one of the following: ]0, 1[, [0, 1[ and [0, 1]. We also show that for k even and 1 ≤ r < 2s, (r, s) = (1, 1) and (r, s) = (3, 2), there is a k-to-1 map from K 2r onto K 2s if and only if k ≥ 2s.

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The core of a graph G is the subgraph GΔ induced by the vertices of maximum degree. We define the deleted core D(G) of G. We extend an earlier sufficient condition due to Hoffman [7] for a graph H to be the core of a Class 2 graph, and thereby provide a stronger sufficient condition. The new sufficient condition is in terms of D(H). We show that in some circumstances our condition is necessary; but it is not necessary in general.

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

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