# Integer Concave Cocirculations and Honeycombs

• Alexander V. Karzanov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3064)

## Abstract

A convex triangular grid is a planar digraph G embedded in the plane so that each bounded face is an equilateral triangle with three edges and their union $${\cal R}$$ forms a convex polygon. A function $$h:E(G)\to{\mathbb R}$$ is called a concave cocirculation if h(e)=g(v)–g(u) for each edge e=(u,v), where g is a concave function on $${\cal R}$$ which is affinely linear within each bounded face of G. Knutson and Tao obtained an integrality result on so-called honeycombs implying that if an integer-valued function on the boundary edges is extendable to a concave cocirculation, then it is extendable to an integer one.

We show a sharper property: for any concave cocirculation h, there exists an integer concave cocirculation h′ satisfying h′(e)=h(e) for each edge e with $$h(e)\in{\mathbb Z}$$ contained in the boundary or in a bounded face where h is integer on all edges.

Also relevant polyhedral and algorithmic results are presented.

### Keywords

Planar graph Discrete convex (concave) function Honeycomb

### AMS Subject Classification

90C10 90C27 05C99

## Preview

### References

1. 1.
Buch, A.S.: The saturation conjecture (after A. Knutson and T. Tao). With an appendix by William Fulton. Enseign. Math (2) 46(1-2), 43–60 (2000)
2. 2.
De Loera, J.A., McAllister, T.B.: Vertices of Gelfand-Tsetlin polytopes, arXiv:math.CO/0309329 (2003)Google Scholar
3. 3.
Karzanov, A.V.: Concave cocirculations in a triangular grid, submitted to Linear Algebra and Appl.Google Scholar
4. 4.
Knutson, A., Tao, T.: The honeycomb model of GLn(C) tensor products I: Proof of the saturation conjecture. J. Amer. Math. Soc. 12(4), 1055–1090 (1999)
5. 5.
Knutson, A., Tao, T., Woodward, C.: The honeycomb model of GLKnutson, A., Tao, T., Woodward, C.: The honeycomb model of GLn(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, Preprint, 2001; to appear in J. Amer. Math. Soc. (2001)Google Scholar