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Keywords:

• k-to-1;
• continuity;
• finite discontinuity;
• graphs;
• complete graphs;
• wiggles;
• 2000 Mathematics Subject Classifications;
• 57M15;
• 05C99

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 onto , which is in fact our main initial emphasis. Second, given two graphs, G and H, and an integer , and considering G and H as subsets of , 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 and . In the cases where n and m have the same parity, and , then we determine exactly when there is a k-to-1 continuous function from onto . 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).