# Color or cover

@article{Izmestiev2015ColorOC, title={Color or cover}, author={Ivan Izmestiev}, journal={arXiv: Combinatorics}, year={2015} }

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can associate a branched cover. This generalizes to a space of germs between two triangulated surfaces. We also discuss relations with Belyi surfaces and with cone-metrics of constant… Expand

#### 4 Citations

Simplicial moves on balanced complexes

- Mathematics
- 2015

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly $(d+1)$-colored) triangulation of a combinatorial $d$-manifold into another balanced triangulation. These… Expand

Sparse Polynomial Systems with many Positive Solutions from Bipartite Simplicial Complexes

- Mathematics, Computer Science
- ArXiv
- 2015

It is shown that all the d-simplices of a Triangulation can be positively decorated if and only if the triangulation is balanced, which allows us to identify, among classical families, monomial supports which admit maximally positive systems, i.e. systems all toric complex solutions of which are real and positive. Expand

Eulerian edge refinements, geodesics, billiards and sphere coloring

- Mathematics, Computer Science
- ArXiv
- 2018

This work constructs some ergodic billiards in 2-balls, where the geodesics bouncing off at the boundary symmetrically and which visit every interior edge exactly once, and tells that every 2-ball can be edge refined using interior edges to become Eulerian if and only if its boundary has length divisible by 3. Expand

A Polyhedral Method for Sparse Systems with Many Positive Solutions

- Computer Science, Mathematics
- SIAM J. Appl. Algebra Geom.
- 2018

We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of posit...

#### References

SHOWING 1-9 OF 9 REFERENCES

Branched Coverings, Triangulations, and 3-Manifolds

- Mathematics
- 2001

A canonical branched covering over each su‰ciently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable… Expand

Projectivities in simplicial complexes and colorings of simple polytopes

- Mathematics
- 2001

Abstract. For each strongly connected finite-dimensional (pure) simplicial complex
$\Delta$ we construct a finite group
$\Pi(\Delta)$, the group of projectivities of
$\Delta$, which is a… Expand

There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

- Mathematics
- 2013

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a… Expand

Graphs on Surfaces and Their Applications

- Mathematics
- 2003

0 Introduction: What is This Book About.- 1 Constellations, Coverings, and Maps.- 2 Dessins d'Enfants.- 3 Introduction to the Matrix Integrals Method.- 4 Geometry of Moduli Spaces of Complex Curves.-… Expand

The nonexistence of colorings

- Mathematics, Computer Science
- J. Comb. Theory, Ser. B
- 1978

This note is to prove the following result about the nonexistence of colorings. Expand