Continuous k‐to‐1 functions between complete graphs whose orders are of a different parity

Abstract

Abstract:A function between graphs is k-to-1 if each point in the codomain 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 consider graphs G and H whose order is of a different parity and determine the even and odd values of k for which there exists a k-to-1 map from G onto H. We first consider k-to-1 maps from K 2r onto K 2s+1 and prove that for 1 ≤ r ≤ s, (r, s) = (1, 1), there is a continuous k-to-1 map for k even if and only if k ≥ 2s and for k odd if and only if k ≥ s o (where s o indicates the next odd integer greater than or equal to s). We then consider k-to-1 maps from K 2s+1 onto K 2s . We show that for 1 ≤ r < s, such a map exists for even