Linear colorings of simplicial complexes and collapsing

Abstract

A vertex coloring of a simplicial complex $Δ$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $Δ$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that $v_1\in F_1\backslash F_2$ and $v_2\in F_2\backslash F_1$. We show that every simplicial complex $Δ$ which is linearly colored with $k$ colors includes a subcomplex $Δ'$ with $k$ vertices such that $Δ'$ is a strong deformation retract of $Δ$. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.