AgraphGisκ-orderedHamiltonian2≤κ≤n,ifforeveryorderedsequenceSofκdistinctverticesofG,thereexistsaHamiltoniancyclethatencountersSinthegivenorder,Inthisarticle,weprovethatifGisagraphonnverticeswithdegreesumofnonadjacentverticesatleastn+3κ-9/2,thenGisκ-orderedHamiltonianforκ=3,4,…，[n/19].Wealsoshowthatthedegreesumboundcanbereducedton+2[κ/2]-2ifκ(G）≥3κ-1/2orδ（G）≥5κ-4.Severalknownresultsaregeneralized.
