A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs
© 2017 Wiley Periodicals, Inc. 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.
Journal of Combinatorial Designs
A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs.
Journal of Combinatorial Designs,
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