Plane curves in an immersed graph in $R^2$

Abstract

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one half.