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Discrete Mathematics
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A vertex coloring of a simplicial complex @D is called a linear coloring if it satisfies the property that for every pair of facets (F"1,F"2) of @D, 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 numberlchr(@D) of @D is defined as the minimum integer k such that @D has a linear coloring with k colors. We show that if @D is a simplicial complex with lchr(@D)=k, then it has a subcomplex @D^' with k vertices such that @D is simple homotopy equivalent to @D^'. As a corollary, we obtain that lchr(@D)=Homdim(@D)+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.