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Paper   IPM / M / 17633
School of Mathematics
  Title:   Cycle algebras and polytopes of matroids
  Author(s):  Sara Saeedi Madani (Joint with T. Roemer)
  Status:   Published
  Journal: J. Pure Appl. Algebra
  Vol.:  228
  Year:  2024
  Pages:   1-23
  Supported by:  IPM
Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

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