Narrow Results

We prove that every spatial embedding of the complete graph K₈ contains at least 3 knotted Hamiltonian cycles, and that every spatial embedding of K_n contains at least 3(n - 1)(n - 2) ⋯ 8 knotted Hamiltonian cycles, for n > 8.

This paper introduces a number of new intrinsically 3-linked graphs through five new constructions. We prove that intrinsic 3-linkedness is not preserved by $\text{Y}\nabla$ moves. We will see that the graph $M$, which is obtained through a $\text{Y}\nabla$ move on ${(PG)}_{\ast }^{\ast }(PG)$, is not intrinsically 3-linked.