On the Hamilton-Waterloo problem with triangle factors and C3x-factors

Abstract

The Hamilton-Waterloo Problem (HWP) in the case of C-m-factors and C-n-factors asks whether K-v, where v is odd (or K-v -F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made either entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. In this paper, we give some general constructions for decompositions and apply them to the case where m = 3 and n = 3x. We settle the problem for odd v, except for a finite number of x values. When v is even, we make significant progress on the problem, although open cases are left. In particular, the difficult case of v even and s = 1 is left open for many situations.