Linear colorings of simplicial complexes and collapsing

A vertex coloring of a simplicial complex Δ is called a linear coloring if it satisfies the property that for every pair of facets (F1, F2) of Δ, there exists no pair of vertices (v1, v2) with the same color such that v1 ∈ F1 {set minus} F2 and v2 ∈ F2 {set minus} F1. The linear chromatic numberlchr (Δ) of Δ is defined as the minimum integer k such that Δ has a linear coloring with k colors. We show that if Δ is a simplicial complex with lchr (Δ) = k, then it has a subcomplex Δ′ with k vertices such that Δ is simple homotopy equivalent to Δ′. As a corollary, we obtain that lchr (Δ) ≥ Homdim (Δ) + 2. We also show in the case of linearly colored simplicial complexes, the usual assignment of a simplicial complex to a multicomplex has an inverse. Finally, we show that the chromatic number of a simple graph is bounded from above by the linear chromatic number of its neighborhood complex. © 2007 Elsevier Inc. All rights reserved.