Realizable winding-number collections for specific nonplanar graphs

Determine the collections of integers realizable as winding numbers w_f(C,v) by almost embeddings into the plane for each of the following graphs: K_5 with one edge deleted, K_{3,3} with one edge deleted, and the graph of a cube or an octahedron.

Background

The paper studies winding numbers associated with an almost embedding f:K→R², where v is a vertex and C is an oriented simple cycle in K\v. For K_4, the authors prove that every integer quadruple of winding numbers whose sum is odd is realizable, showing that parity is the only relation in that case.

The paper formulates a broader realization problem for arbitrary finite graphs and then identifies three unresolved instances: K_5 with an edge deleted, K_{3,3} with an edge deleted, and the graphs of a cube and an octahedron. The cited results provide partial information for some of these graphs but do not determine all realizable winding-number collections.

References

The answer to the open problem in the following interesting cases is unknown: \begin{itemize} \item $K$ is obtained from $K_5$ by deleting an edge (cf. Theorem 1); \item $K$ is obtained from $K_{3, 3}$ by deleting an edge (cf. \S5, 5.7); \item $K$ is the graph of a cube or an octahedron (cf. \S5, 5.8). \end{itemize}

— On winding numbers of almost embeddings of $K_4$ in the plane  (2501.15642 - Alkin et al., 26 Jan 2025) in Section 5, “Discussion and open problem,” immediately following Open Problem; page number unavailable