Realizable winding-number collections for unresolved graph classes

Determine the collections of winding numbers realizable by almost embeddings into the plane for each of the following finite graphs: the graph obtained from $K_5$ by deleting an edge, the graph obtained from $K_{3,3}$ by deleting an edge, and the graph of a cube or an octahedron.

Background

The paper asks, for a fixed finite graph KK, which collections of integers wf(C,v)w_f(C,v)—indexed by vertices vv and oriented simple cycles CC not containing vv—can arise from almost embeddings f:K→R2f:K\to\mathbb{R}^2. The paper completely resolves this question for proper subgraphs of K4K_4 and for K4K_4 itself, proving that the parity condition on the four winding numbers is the only relation in the K4K_4 case.

The authors explicitly state that the corresponding realizability classification remains unknown for three further graph classes: K5K_5 with one edge deleted, K3,3K_{3,3} with one edge deleted, and the graphs of a cube or an octahedron.

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 4, “Discussion and open problem” (following Open Problem, near the end of the section)