As　a　natural　generalization　of　graph　coloring,　Vince　introduced　the　star　chromatic　number　of　a　graph　G　and　denoted　it　by　χ*(G).　Later,　Zhu　called　it　circular　chromatic　number　and　denoted　it　by　χc(G).　Let　χ(G)　be　the　chromatic　number　of　G.　In　this　paper,　it　is　shown　that　if　the　complement　of　G　is　non-hamiltonian,　then　χc(G)　=χ(G).　Denote　by　M(G)　the　Mycielski　graph　of　G.　Recursively　define　Mm(G)　=　M(M　m-1(G)).　It　was　conjectured　that　if　m　≤　n　-　2,　then　χc(Mm(Kn))　=　χ(Mm(K　n)).　Suppose　that　G　is　a　graph　on　n　vertices.　We　prove　that　if　χ(G)　≥　n+s/2,　then　χc(M(G))　=　χ(M(G)).　Let　S　be　the　set　of　vertices　of　degree　n-1　in　G.　It　is　proved　that　if　|S|　≥　3,　then　χc(M(G))　=　χ(M(G)),　and　if　|S|　≥　5,　then　χc(M2(G))　=　χ(M2(G)),　which　implies　the　known　results　of　Chang,　Huang,　and　Zhu　that　if　n≥3,　χc(M(Kn))=χ(M(Kn)),　and　if　n≥5,　then　χc(M3(Kn))=χ(M2(K　n)).