Wiggles and Finitely Discontinuous k-to-1 Functions Between Graphs

The graphs we shall consider are topological graphs that is they lie in R3 and each edge is homeomorphic to [0, 1]. If a graph is simple, that is it has no loops or multiple edges, then each edge may be taken to be a straight line joining the two vertices at the ends of the edge. A function is k-to-1 if each point in the codomain has precisely k preimages in the domain. Given two graphs G and H, and an integer k ≥ 1, Jo Heath proved the surprising result that there exists a finitely discontinuous k-to-1 function f from G onto H if and only if |E(G)| − |V (G)| ≤ k (|E(H)| − |V (H)|) if k ≥ 3 , and |E(G)| − |V (G)| = 2 (|E(H)| − |V (H)|) if k = 2 . Such functions often involve a limit construction, which we call a wiggle. In this talk, I shall discuss a simple formula (related to Jo Heath’s result) which counts the number of wiggles. I shall also discuss the special case when the finitely discontinuous function f can actually be chosen to be continuous. Much of the talk will be joint work from the past with Jo Heath, or current work with my student, John Gauci.