By　Smith＇s　theorem,　if　a　cubic　graph　has　a　Hamiltonian　cycle,　then　it　has　a　second　Hamiltonian　cycle.　Thomason　[＇Hamilton　cycles　and　uniquely　edge-colourable　graphs＇,　Ann.　Discrete　Math.　3　(1978),　259-268]　gave　a　simple　algorithm　to　find　the　second　cycle.　Thomassen　[private　communication]　observed　that　if　there　exists　a　polynomially　bounded　algorithm　for　finding　a　second　Hamiltonian　cycle　in　a　cubic　cyclically　4-edge　connected　graph,　then　there　exists　a　polynomially　bounded　algorithm　for　finding　a　second　Hamiltonian　cycle　in　any　cubic　graph.　In　this　paper　we　present　a　class　of　cyclically　4-edge　connected　cubic　bipartite　graphs　with　vertices　such　that　Thomason＇s　algorithm　takes　steps　to　find　a　second　Hamiltonian　cycle　in.　©　2018　Australian　Mathematical　Publishing　Association　Inc.