• Corpus ID: 244130347

Criteria for integer and modulo 2 embeddability of graphs to surfaces

@inproceedings{Bikeev2020CriteriaFI,
  title={Criteria for integer and modulo 2 embeddability of graphs to surfaces},
  author={Arthur Bikeev},
  year={2020}
}
The study of graph drawings on 2-surfaces is an active area of mathematical research. Surveying these studies is beyond the scope of the present paper; see Remark 1.3 for results most closely related to ours. Our main results are criteria for Z2-embeddability and Z-embeddability (see definitions below) of graphs to surfaces (Theorems 1.1 and 1.4). See Remarks 1.2 and 1.5 for applications, comments and relations to other results. In this paper we use the following conventions and notations. Let… 
3 Citations

Figures from this paper

To the Kühnel conjecture on embeddability of k-complexes in 2k-manifolds

. The classical Heawood inequality states that if the complete graph K n on n vertices is embeddable in the sphere with g handles, then g > ( n − 3)( n − 4) 12 . A higher-dimensional analogue of the

A short exposition of the Patak-Tancer theorem on non-embeddability of k-complexes in 2k-manifolds

In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short

Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

TLDR
This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

References

SHOWING 1-10 OF 29 REFERENCES

Invariants of Graph Drawings in the Plane

TLDR
A simplified exposition of some classical and modern results on graph drawings in the plane, as an introduction to starting ideas of algebraic topology motivated by algorithmic, combinatorial and geometric problems.

Embeddings of k-complexes in 2k-manifolds and minimum rank of partial symmetric matrices

TLDR
This work proves that for k ≥ 3 odd K embeds into M if and only if there are a skew-symmetric n × n-matrix A with Z-entries whose rank over Q does not exceed rkHk(M ;Z), and a collection of orientations on k-faces of K such that for any nonadjacent k- faces σ, τ the element Aσ,τ equals to the algebraic intersection of fσ and fτ.

On the rank of Z_2-matrices with free entries on the diagonal

TLDR
It is proved that for each non-negative integer k there is a polynomial in n algorithm deciding whether R(M) ≤ k (whose complexity may depend on k) and that there is an algorithm with the complexity of O(n) calculating for an arbitrary matrix M ∈ Mn(Z2) a number k such that k/2 ≤ R( M) ≥ k.

The disjoint paths problem in quadratic time

Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

TLDR
This paper uses a counterexample to an extension of the unified Hanani-Tutte theorem on the torus to find a graph of genus $5$ and its drawing on the orientable surface of genus £4 with every pair of independent edges crossing an even number of times.

Z_2-genus of graphs and minimum rank of partial symmetric matrices

TLDR
The $\mathbb{Z}_2$-genus of a graph is expressed using the minimum rank of partial symmetric matrices over $\Mathbb{ Z}_ 2$; a problem that might be of independent interest.

Algebraic topology from geometric viewpoint

This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology

Embeddings of k-complexes into 2k-manifolds

TLDR
The $\mathbb Z_2$-reduction of the obstruction shows how to obtain a non-trivial upper bound for the Kuhnel problem: determine the smallest $n$ so that $\Delta_n^{(k)}$ does not embed into $M$.

Symmetric and alternate matrices in an arbitrary field. I

The elementary theorems of the classical treatment of symmetric and alternate matrices may be shown, without change in the proofs, to hold for matrices whose elements are in any field of