Wiggles and finitely discontinuous 𝘬-to-1 functions between graphs

Abstract

A function between graphs is k-to-1 if each point in the codomain has precisely k preimages in the domain. In this article, we approach the topic of continuous, or finitely discontinuous, k-to-1 functions between graphs from three different points of view. Harrold (Duke Math J 5 (1939), 789-793) showed that there is no 2-to-1 continuous function from a closed interval onto a circle (i.e., from K2 onto C3). In the first part of this article, we describe all 3-to-1 continuous functions from an edge onto a cycle. Such a description is just one step away from a description of all 3-to-1 continuous functions from R onto R, which is in fact our main initial emphasis. Second, given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of R3, Jo Heath gave a simple criterion for the existence of a finitely discontinuous k-to-1 function from G onto H. Such functions often involve a limiting construction which we call a wiggle. In the second part of this article, we give a simple formula (related to Jo Heath's construction) which counts the number of wiggles. The question of whether there is a continuous k-to-1 function (i.e., a k-to-1 map in the usual topological sense) from G onto H is more complicated. In the third part of this article, we consider complete graphs Kn and Km. In the cases where n and m have the same parity, and n≤m, then we determine exactly when there is a k-to-1 continuous function from Kn onto Km. Other cases are considered elsewhere. (J. K. Dugdale, S. Fiorini, A. J. W. Hilton, J. B. Gauci, Discrete Math 310 (2010), 330–346 and J. B. Gauci, A. J. W. Hilton, J Graph Theory 65 (2010), 35–60).