On Connected m-HPK(n1,n2,n3,n4)

Abstract

We definem-HPK(n1,n2,n3,n4)[Kt]-residual graphs in whichHPKis a hyperplane complete graph. We extend P. Erdös, F. Harary, and M. Klawe's definition of plane complete residual graph to hyperplane and obtain the hyperplane complete residual graph. Further, we obtain the minimum order ofHPK(n1,n2,n3,n4)[Kt]-residual graphs andm-HPK(n1,n2,n3,n4)[Kt]-residual graphs. In addition, we obtain a unique minimalHPK(n1,n2,n3,n4)[Kt]-residual graphs and a unique minimalm-HPK(n1,n2,n3,n4)[Kt]-residual graphs.