Towards a short proof of the Fulek–Kynčl criterion for modulo 2 embeddability of graphs to surfaces

12/22/2020

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by   Arthur Bikeev, et al.

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A connected graph K has a modulo 2 embedding to the sphere with g handles if and only if there is a general position PL map f of K in the plane and a symmetric square matrix A of size |E(K)| with modulo 2 entries and zeros on the diagonal such that rk A ≤ 2g and A_σ, τ = |fσ∩ fτ| mod 2 for any non-adjacent edges σ, τ. This is essentially proved by R. Fulek and J. Kynčl. The main of results of this note is an alternative proof of the (=>) part.