Continuous k -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 topologi...

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Bibliographic Details
Published in:Discrete mathematics 2010-01, Vol.310 (2), p.330-346
Main Authors: Dugdale, J. Keith, Fiorini, Stanley, Hilton, Anthony J.W., Gauci, John Baptist
Format: Article
Language:English
Subjects:
[formula omitted]-to-1
Algebra
Applied sciences
Combinatorics
Combinatorics. Ordered structures
Complete graphs
Computer science
control theory
systems
Exact sciences and technology
Functions
Information retrieval. Graph
Mathematical analysis
Mathematics
Real functions
Sciences and techniques of general use
Theoretical computing
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Summary: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 .
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2008.11.036