It reminds me of a variation of the bridges of konigsberg problem where in this diagram as presented, it has an even number of vertices and edges, and an implie

It reminds me of a variation of the bridges of konigsberg problem where in this diagram as presented, it has an even number of vertices and edges, and an implied even number of faces, but the vertices have an odd number of edges, which breaks the need for the number of edges per vertex to be even to do a hamiltonian path. (not a mathematician, can't remember this rule specifically) I'd wonder if there is some topology theorem that generalizes hamiltonian paths into higher dimensions, where the difference between a graph and a solid is whether it has a hamiltonian path. e.g. this shape is "impossible" as represented because of the lack of a hamiltonian path.