Continuous kk-to-1 functions between complete graphs of even order

A function between graphs is kk-to-1 if each point in the co-domain has precisely kk pre-images in the domain. Given two graphs, GG and HH, and an integer k≥1k≥1, and considering GG and HH as subsets of R3R3, there may or may not be a kk-to-1 continuous function (i.e. a kk-to-1 map in the usual topological sense) from GG onto HH. In this paper we review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous kk-to-1 functions between two intervals, each of which is one of the following: ]0,1[]0,1[, [0,1[[0,1[ and [0,1][0,1]. We also show that for kk even and 1≤r<2s1≤r<2s, (r,s)≠(1,1)(r,s)≠(1,1) and (r,s)≠(3,2)(r,s)≠(3,2), there is a kk-to-1 map from K2rK2r onto K2sK2s if and only if k≥2sk≥2s.