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 (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

Colinear Coloring on Graphs

“…In this section we define the colinear coloring of a graph G, and we prove some properties of the colinear coloring of G. It is worth noting that these properties have been also proved for the linear coloring of the neighborhood complex N (G) in [5].…”

“…Motivated by the definition of linear coloring on simplicial complexes associated to graphs, first introduced by Civan and Yalçin [5] in the context of algebraic topology, we studied linear colorings on simplicial complexes which can be represented by a graph. In particular, we studied the linear coloring problem on a simplicial complex, namely independence complex I(G) of a graph G. The independence complex I(G) of a graph G can always be represented by a graph and, more specifically, is identical to the complement graph G of the graph G; indeed, the facets of I(G) are exactly the maximal cliques of G. The outcome of this study was the definition of the colinear coloring of a graph G; the colinear coloring of a graph G is a coloring of G such that for any set of vertices taking the same color, the collection of their clique sets can be linearly ordered by inclusion.…”

“…In particular, we studied the linear coloring problem on a simplicial complex, namely independence complex I(G) of a graph G. The independence complex I(G) of a graph G can always be represented by a graph and, more specifically, is identical to the complement graph G of the graph G; indeed, the facets of I(G) are exactly the maximal cliques of G. The outcome of this study was the definition of the colinear coloring of a graph G; the colinear coloring of a graph G is a coloring of G such that for any set of vertices taking the same color, the collection of their clique sets can be linearly ordered by inclusion. Note that, the two definitions cannot always be considered as identical since not in all cases a simplicial complex can be represented by a graph; such an example is the neighborhood complex N (G) of a graph G. Recently, Civan and Yalçin [5] studied the linear coloring of the neighborhood complex N (G) of a graph G and proved that the linear chromatic number of N (G) gives an upper bound for the chromatic number χ(G) of the graph G. This approach lies in a general framework met in algebraic topology.…”

Harmonious Coloring on Subclasses of Colinear Graphs

“…Motivated by the definition of linear coloring on simplicial complexes associated to graphs, first introduced in the context of algebraic topology [8], we recently introduced the colinear coloring on graphs [17].…”

Dismantlability of weakly systolic complexes and applications

“…We now consider the analogue of dismantlability for a simplicial complex X investigated in the papers [CY07,Mat08]. A vertex v of X is LC-removable if lk(v, X) is a cone.…”