We prove that the neighborhood complex $\N(G)$, the box complex $\B(G)$, the homomorphism complex $\Hom(K_2,G)$and the Lov\'{a}sz complex $\L(G)$ have the same simple $\Z_2$-homotopy type in the sense of Whitehead. We show that these graph complexes are simple $\Z_2$-universal.

Keywords:

graph complexes, simple $\Z_2$-homotopy, universality graph complexes, simple $\Z_2$-homotopy, universality

MSC Classifications:

57Q10, 05C10, 55P10 show english descriptions Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]
Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
Homotopy equivalences 57Q10 - Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc. [See also 19B28]
05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
55P10 - Homotopy equivalences

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