Continuous k‐to‐1 functions between complete graphs whose orders are of a different parity
John Baptist Gauci, Anthony J. W. Hilton · Journal of Graph Theory · 2009
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