Lattice of Integer Flows and Poset of Strongly Connected Orientations

Details Lattice of Integer Flows and Poset of Strongly Connected Orientations Lattice of Integer Flows and Poset of Strongly Connected Orientations

2010-07-15

arxiv.org/abs/1007.2456v1

Mathematics

Combinatorics

16 pages

Scientific paper

We show that the Voronoi cells of the lattice of integer flows of a finite connected graph $G$ in the quadratic vector space of real valued flows have the following very precise combinatorics: the face poset of a Voronoi cell is isomorphic to the poset of strongly connected orientations of subgraphs of $G$. This confirms a recent conjecture of Caporaso and Viviani {Torelli Theorem For Graphs and Tropical Curves, Duke Math. J. 153(1) (2010), 129-171}. We also prove an analogue theorem for the lattice of integer cuts.

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