As　a　natural　generalization　of　graph　coloring,　Vince　introduced　the　star　chromatic　number　of　a　graph　G　and　denoted　it　by　chi*(G).　Later,　Zhu　called　it　circular　chromatic　number　and　denoted　it　by　chi(c)(G).　Let　chi(G)　be　the　chromatic　number　of　G.　In　this　paper,　it　is　shown　that　if　the　complement　of　G　is　non-hamiltonian,　then　chi(c)(G)=chi(G).　Denote　by　M(G)　the　Mycielski　graph　of　G.　Recursively　define　M-m(G)=M(Mm-1(G)).　It　was　conjectured　that　if　mless　than　or　equal　ton-2,　then　chi(c)(M-m(K-n))=chi(M-m(K-n)).　Suppose　that　G　is　a　graph　on　n　vertices.　We　prove　that　if　chi(G)　greater　than　or　equal　to　n+3/2,　then　chi(c)(M(G))=chi(M(G)).　Let　S　be　the　set　of　vertices　of　degree　n-1　in　G.　It　is　proved　that　if　|S|greater　than　or　equal　to　3,　then　chi(c)(M(G))=chi(M(G)),　and　if　|S|greater　than　or　equal　to　5,　then　chi(c)(M-2(G))=chi(M-2(G)),　which　implies　the　known　results　of　Chang,　Huang,　and　Zhu　that　if　ngreater　than　or　equal　to3,　chi(c)(M(K-n))=chi(M(K-n)),　and　if　ngreater　than　or　equal　to5,　then　chi(c)(M-2(K-n))=chi(M-2(K-n)).