Menu
Given two non-isomorphic bipartite 2-factors \( F_1 \) and \( F_2 \) of order \( 4n \), the Bipartite Hamilton-Waterloo Problem (BHWP) asks for a 2-factorization of \( K_{2n,2n} \) into \( \alpha \) copies of \( F_1 \) and \( \beta \) copies of \( F_2 \), where \( \alpha + \beta = n \) and \( \alpha, \beta \geq 1 \). We show that the BHWP has a solution when \( F_1 \) is a refinement of \( F_2 \), where no component of \( F_1 \) is a \( C_4 \) or \( C_6 \), except possibly when \( \alpha = 1 \) and either (i) \( F_2 \) is a \( C_4 \)-factor or (ii) \( F_2 \) has more than one \( C_4 \) with all other components of an order \( r \equiv 0 \pmod{4} > 4 \) or (iii) \( F_2 \) has components with an order \( r \equiv 2 \pmod{4} \), when \( n \) is even. We also show that there does not exist a factorization of \( K_{6,6} \) into a single 12-cycle and two \( C_4 \)-factors.