On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

Abstract

The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2 -factor F 1 and s copies of a 2 -factor F 2 such that r + s = ⌊( v − 1) / 2⌋ . If F 1 consists of m -cycles and F 2 consists of n cycles, we say that a solution to ( m , n ) - HWP( v ; r , s ) exists. The goal is to find a decomposition for every possible pair ( r , s ) . In this paper, we show that for odd x and y , there is a solution to (2 k x , y ) - HWP( v m ; r , s ) if gcd ( x , y ) ≥ 3 , m ≥ 3 , and both x and y divide v , except possibly when 1 ∈ { r , s } .