# 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

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

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. 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

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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

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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.

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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.

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