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

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 ℝ3, 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 K2r onto K2s+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 K2s+1 onto K2s. We show that for 1 ≤ r< s, such a map exists for even values of k if and only if k≥2s. We also prove that whatever the values of r and s are, no such k-to-1 map exists for odd values of k. To conclude, we give all triples (n, k, m) for which there is a k-to-1 map from Kn onto Km in the case when n≤m.