2025/01/27 by Ákos G. Horváth, Horváth, Ákos G., István Prok +1
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research
paper · pdf · doi:10.48550/arxiv.2501.16121
This article aims to study the class of strongly self-dual polytopes (ssd-polytopes for short), defined in a paper by Lovász \citelovasz. He described a series of such polytopes (called L-type polytopes), which he used to solve a combinatorial problem. From a geometrical point of view, there are interesting questions: what additional elements of this class exist, and are there any with a different structure from the L-type ones? We show that in dimension three, one of their faces defines L-type polyhedra. Illustrating the algorithm of the proof, we present an ssd-polytope of 23 vertices whose combinatorial structure differ from those of L-type ones. Finally, with an elementary discussion, we prove that for fewer than nine vertices, there are only fifth one ssd-polyhedra, four of them can be constructed by Lovász's method, and we can find the fifth one with "the diameter gradient flow algorithm" of Katz, Memoli and Wang \citekatz-memoli-wang.