Continuous -to-1 functions between complete graphs of even order

A function between graphs is k -to-1 if each point in the co-domain 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 R 3 , 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 review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous k -to-1 functions between two intervals, each of which is one of the following: ] 0 , 1 [ , [ 0 , 1 [ and [ 0 , 1 ] . We also show that for k even and 1 ≤ r < 2 s , ( r , s ) ≠ ( 1 , 1 ) and ( r , s ) ≠ ( 3 , 2 ) , there is a k -to-1 map from K 2 r onto K 2 s if and only if k ≥ 2 s .