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On the Lattice of Cycles of an Undirected Graph

Algebra & Discrete Mathematics

Speaker: Anastasia Chavez, UC Davis
Location: Zoom
Start time: Thu, Mar 4 2021, 9:30AM

The structure of the cycles of a graph is a well-studied topic with deceivingly simple open questions. For example, the Double Cover Conjecture claims that there is a family of cycles such that every edge of a bridgeless graph is contained in exactly two cycles. A linear-algebraic view of the problem is a natural point of view. Considering every cycle as an indicator vector, we explore the lattice $L_G$ generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. This linear algebraic perspective has been applied historically to model combinatorial problems on directed graphs with great success (in some textbooks), whereas the undirected edge case less so. In this talk, we present structural results for the lattice $L_G$, including explicit formulas for its dimension and determinant, and we present a quadratic time algorithm to construct lattice bases, using only cycles as generators.

This is joint work with Gennadiy Averkov, Jesus De Loera, and Bryan Gillespie. Our paper is available on the ArXiv.


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