Intrinsic Knotting of Almost Complete Partite Graphs

Abstract

Let $G$ be a complete $p$-partite graph with 2 edges removed, $p ≥ 7$, which is intrinsically knotted. Let $J$ represent any graph obtained from $G$ by a finite sequence of $∆-Y$ exchanges and/or vertex expansions. In the present paper, we show that the removal of any vertex of $J$ and all edges incident to that vertex produces an intrinsically linked graph. This result offers more intrinsically knotted graphs which hold for the conjecture presented in Adams' book (Adams C. The Knot Book. New York: W. H. Freeman and Company, 1994), that is, the removal of any vertex from an intrinsically knotted graph yields an intrinsically linked graph.