## ON THE GRAPH COLORING POLYTOPE

Citations: | 1 - 1 self |

### BibTeX

@MISC{Palubeckis_onthe,

author = {Gintaras Palubeckis},

title = {ON THE GRAPH COLORING POLYTOPE},

year = {}

}

### OpenURL

### Abstract

Abstract. The graph coloring problem consists in assigning colors to the vertices of agiven graph Gsuch that no two adjacent vertices receive the same color and the number of used colors is as small as possible. In this paper, we investigate the graph coloring polytope P (G) defined as the convex hull of feasible solutions to the binary programming formulation of the problem. We remark that P (G) coincides with the stable set polytope of a graph constructed from the complement ¯ G of G. We derive facet-defining inequalities for P (G) from independent sets, odd holes, odd anti-holes and odd wheels in ¯ G. Key words: polyhedral combinatorics, graph coloring, polytopes, facets. 1.

### Citations

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Combinatorial optimization: Polyhedra and Efficiency
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Citation Context ...polytope Pstable(G) of a graph G is the convex hull of the incidence vectors of the independent (stable) setsG. Palubeckis in G. The basic concepts of polyhedral theory can be found, for example, in =-=[15]-=-. 2. Facets of the polytope In this section, we exhibit a few classes of facetdefining inequalities for P (G). We start by pointing out a one-to-one correspondence between colorings of G and admissibl... |

512 |
Optimization by Simulated Annealing: An Experimental Evaluation
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- 1991
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Citation Context ...nimum number χ(G) of colors is called the chromatic number of the graph G. There exist many approaches for the graph coloring problem. These approaches include both exact [2, 4, 10, 16] and heuristic =-=[1, 7, 8, 9]-=- solution methods. In the development of algorithms for graph coloring, various integer programming formulations of the problem could be used. Several such formulations, each involving binary variable... |

350 |
New Methods to Color Vertices of a Graph
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(Show Context)
Citation Context ...as small as possible. This minimum number χ(G) of colors is called the chromatic number of the graph G. There exist many approaches for the graph coloring problem. These approaches include both exact =-=[2, 4, 10, 16]-=- and heuristic [1, 7, 8, 9] solution methods. In the development of algorithms for graph coloring, various integer programming formulations of the problem could be used. Several such formulations, eac... |

132 |
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- Hertz, Werra
- 1987
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Citation Context ...nimum number χ(G) of colors is called the chromatic number of the graph G. There exist many approaches for the graph coloring problem. These approaches include both exact [2, 4, 10, 16] and heuristic =-=[1, 7, 8, 9]-=- solution methods. In the development of algorithms for graph coloring, various integer programming formulations of the problem could be used. Several such formulations, each involving binary variable... |

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100 |
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Citation Context ...k) ∈ Ē′′ i . The maximality of I implies the existence of a vertex j ∈ I such that (j, k) ∈ Ē. However, (i, j, k) ∈ T and hence (tij, tik) ̸∈ EH, a contradiction to the above assumption. As proved in =-=[13]-=-, the inequality ∑ i∈V ′ xi � 1 for the vertex set V ′ of an inclusion-wise maximal clique in a graph G defines a facet of Pstable(G). Applying this result to HG and using Proposition 1, we conclude t... |

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39 |
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Citation Context ...raph of HG induced by the vertices tci, i ∈ Nc(W ), is an odd anti-hole A = (VA, EA). For its vertex set VA, the inequality ∑ tci∈VA xtci � 2 (9) defines a facet of the stable set polytope Pstable(A) =-=[12]-=-. This fact establishes the validity of (8). Now suppose that the stated conditions hold for W . Similarly to the proof of Theorem 2, we construct the required collection of linearly independent incid... |

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Citation Context ...as small as possible. This minimum number χ(G) of colors is called the chromatic number of the graph G. There exist many approaches for the graph coloring problem. These approaches include both exact =-=[2, 4, 10, 16]-=- and heuristic [1, 7, 8, 9] solution methods. In the development of algorithms for graph coloring, various integer programming formulations of the problem could be used. Several such formulations, eac... |

8 |
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Citation Context ... an integer program with a variable for each possible color and vertex [5, 11], a model relating acyclic orientations of a graph to its chromatic number [6], a model with vertices representing colors =-=[3]-=-, and a formulation based on star partitioning of the complement of a given graph [14]. Let ¯ G = (V, Ē) denote the complement of the graph G = (V, E) of order n = |V |. Assume (which is not restricti... |

4 |
A polyhedral approach for graph coloring
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Citation Context ... could be used. Several such formulations, each involving binary variables, have been proposed: independent set formulation [10], an integer program with a variable for each possible color and vertex =-=[5, 11]-=-, a model relating acyclic orientations of a graph to its chromatic number [6], a model with vertices representing colors [3], and a formulation based on star partitioning of the complement of a given... |

2 |
New 0-1 integer formulations of the graph coloring problem
- Figueiredo, Barbosa, et al.
- 2002
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Citation Context ...een proposed: independent set formulation [10], an integer program with a variable for each possible color and vertex [5, 11], a model relating acyclic orientations of a graph to its chromatic number =-=[6]-=-, a model with vertices representing colors [3], and a formulation based on star partitioning of the complement of a given graph [14]. Let ¯ G = (V, Ē) denote the complement of the graph G = (V, E) of... |

2 |
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Citation Context ...el relating acyclic orientations of a graph to its chromatic number [6], a model with vertices representing colors [3], and a formulation based on star partitioning of the complement of a given graph =-=[14]-=-. Let ¯ G = (V, Ē) denote the complement of the graph G = (V, E) of order n = |V |. Assume (which is not restrictive) that each connected component of ¯G has order greater than two. Letting V be a set... |

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Citation Context ... could be used. Several such formulations, each involving binary variables, have been proposed: independent set formulation [10], an integer program with a variable for each possible color and vertex =-=[5, 11]-=-, a model relating acyclic orientations of a graph to its chromatic number [6], a model with vertices representing colors [3], and a formulation based on star partitioning of the complement of a given... |