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

Abstract

A function between graphs is k-to-1 if each point in the co-domain has precisely k pre-images 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) 6= (1, 1) and (r, s) 6= (3, 2), there is a k-to-1 map from K2r onto K2s if and only if k ≥ 2s.