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 ( F 1 , F 2 ) of Δ, there exists no pair of vertices ( v 1 , v 2 ) with the same color such that v 1 ∈ F 1 ∖ F 2 and v 2 ∈ F 2 ∖ F 1 . The linear chromatic number lchr ( 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.