Cycle space of graphs of polytopes
Guillermo Pineda-Villavicencio

TL;DR
This paper proves that the cycle space of graphs of polytopes is generated by 2-face bounding cycles, offering a homology-free proof and confirming bipartiteness equivalence between the polytope graph and its 2-faces.
Contribution
It provides a new proof of a known result about cycle spaces of polytope graphs, avoiding homological methods and confirming bipartiteness conditions.
Findings
Cycle space of polytope graphs is generated by 2-face cycles
Graphs of polytopes are bipartite iff all 2-face graphs are bipartite
Homology-free proof simplifies understanding of polytope graph structure
Abstract
It is folklore that the cycle space of graphs of polytopes is generated by the cycles bounding the 2-faces. We provide a proof of this result that bypass homological arguments, which seem to be the most widely known proof. As a corollary, we obtain a result of Blind & Blind (1994) stating that graphs of polytopes are bipartite if and only if graphs of every 2-face are bipartite.
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Combinatorial Mathematics · Commutative Algebra and Its Applications
