Ah, okay, now I understand why you met icosidodecahedron, nice, thanks. And nice theorem. And even looks kinda related to sphere labeling, but not straightforwardly.
I don't believe Aristotle ever wrote "regular". Even thou, today, "platonic solid" is synonym with regular finite convex simple euclidean polyhedron, Plato himself only said every face should have the same number of sides and every vertex the same number of faces.
Oh, btw, Bjorn, have you seen this conjecture - web.archive.org/web/20200113032547/www.openproblemgarden.org/op/unit_vector_flows - about sphere? Probably, everything that I'll write here is not related to your research, but who knows. The conjecture about the nowhere-zero sphere labeling is related to nowhere-zero 5-flow conjecture on bridgeless graphs (or snarks, actually). What I've found out, is that icosidodecahedron is the minimal configuration which can't be labeled with numbers +-1, +-2 and +-3, we'll need +-4. Actually, icosidodecahedron is Petersen graph in disguise, if we consider this sphere labeling conjecture. It's easier to see this , if we look at dodecadodecahedron instead. (Would be nice to find bigger labelings that require +-4 labels, but haven't found them yet) (nowhere-zero 5-flow conjecture is also highly related or connected or reducible in some sense to Petersen graph) So, what I was wondering about - did you meet Petersen graph in your research anywhere?
Ah, okay, now I understand why you met icosidodecahedron, nice, thanks. And nice theorem. And even looks kinda related to sphere labeling, but not straightforwardly.
I don't believe Aristotle ever wrote "regular". Even thou, today, "platonic solid" is synonym with regular finite convex simple euclidean polyhedron, Plato himself only said every face should have the same number of sides and every vertex the same number of faces.
Oh, btw, Bjorn, have you seen this conjecture - web.archive.org/web/20200113032547/www.openproblemgarden.org/op/unit_vector_flows - about sphere? Probably, everything that I'll write here is not related to your research, but who knows.
The conjecture about the nowhere-zero sphere labeling is related to nowhere-zero 5-flow conjecture on bridgeless graphs (or snarks, actually). What I've found out, is that icosidodecahedron is the minimal configuration which can't be labeled with numbers +-1, +-2 and +-3, we'll need +-4. Actually, icosidodecahedron is Petersen graph in disguise, if we consider this sphere labeling conjecture. It's easier to see this , if we look at dodecadodecahedron instead. (Would be nice to find bigger labelings that require +-4 labels, but haven't found them yet) (nowhere-zero 5-flow conjecture is also highly related or connected or reducible in some sense to Petersen graph)
So, what I was wondering about - did you meet Petersen graph in your research anywhere?