On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors. Issue 2 (29th May 2012) - Bryant, Darryn; Danziger, Peter; Dean, Matthew
Abstract Given two 2‐regular graphs F1 and F2, both of order n, the Hamilton‐Waterloo Problem for F1 and F2 asks for a factorization of the complete graph <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n44wbgf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />Kn into α1 copies of F1, α2 copies of F2, and a 1‐factor if n is even, for all nonnegative integers α1 and α2 satisfying <inline-graphic mimetype="image" xlink:href="ark:/27927/pgg1n44wbdb" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />α1+α2=⌊n−12⌋. We settle the Hamilton‐Waterloo Problem for all bipartite 2‐regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2‐regular graph of the same order.