Cycle space of graphs of polytopes

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.