A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs
Document Type
Article
Publication Date
10-2017
Department
Department of Mathematical Sciences
Abstract
The Hamilton–Waterloo problem asks for which s and r the complete graph Kn can be decomposed into s copies of a given 2-factor F1 and r copies of a given 2-factor F2 (and one copy of a 1-factor if n is even). In this paper, we generalize the problem to complete equipartite graphs K(n:m) and show that K(xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm; and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s,r,≠1,gcd(x,z)=gcd(y,z) =1,and xyz≠ 0(mod4). We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton–Waterloo problem for complete graphs.
Publication Title
Journal of Combinatorial Designs
Recommended Citation
Keranen, M. S.,
&
Pastine, A.
(2017).
A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs.
Journal of Combinatorial Designs,
25(10), 431-468.
http://doi.org/10.1002/jcd.21560
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/3715
Publisher's Statement
© 2017 Wiley Periodicals, Inc. Publisher’s version of record: https://doi.org/10.1002/jcd.21560