The representatives formulation for the vertex coloring problem is revisited to remove symmetry a... more The representatives formulation for the vertex coloring problem is revisited to remove symmetry and new versions of facets derived from substructures of the graph are presented. In addition, a new class of facets is derived from independent sets of the graph. Finally, a comparison with the independent sets formulation is provided.
The Seventh European Conference on Combinatorics, Graph Theory and Applications, 2013
A b-coloring of a graph is a proper coloring of its vertices such that every color class contains... more A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b(G) such that the graph has a b-coloring with b(G) colors. This metric is upper bounded by the largest integer m(G) for which G has at least m(G) vertices with degree at least m(G)−1.
A b-coloring of a graph is a coloring of its vertices such that every color class contains a vert... more A b-coloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. We show how to compute in polynomial time the b-chromatic number of a graph of girth at least 9. This improves the seminal result of Irving and Manlove on trees. *
O número cromático fracionário ( ) F G χ de um grafo G é um conhecido limite inferior para seu nú... more O número cromático fracionário ( ) F G χ de um grafo G é um conhecido limite inferior para seu número cromático ( ) G χ . Experimentos relatados na literatura mostram que usar ( ) F G χ , em lugar do tamanho da clique máxima, pode ser muito mais eficiente para orientar a busca em um algoritmo tipo branch-and-bound para determinação de ( ) G χ . Uma dificuldade, porém, é tratar o modelo linear conhecido para ( ) F G χ , o qual apresenta um número exponencial de variáveis e demanda um caro processo de geração de colunas. Neste trabalho, examinamos uma formulação alternativa para obter um limite inferior para ( ) F G χ que possui um número de variáveis linear no tamanho do grafo, porém um número exponencial de restrições. Utilizamos o método de planos-de-corte para lidar com esse inconveniente. Algumas heurísticas de separação são propostas, e experimentos computacionais mostram que valores muito próximos de ( ) F G χ , em muitos casos iguais, são encontrados em tempo inferior à implementação com geração de colunas.
A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex t... more A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G)χb(G) is the maximum number of colours used in a b-colouring of G . If m(G)m(G) is the largest integer k such that G has at least k vertices with degree at least k−1k−1, then we know that χb(G)⩽m(G)χb(G)⩽m(G). Irving and Manlove [Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, 91 (1999), pages 127–141] prove that, if T is a tree, then the b-chromatic number of T is at least m(T)−1m(T)−1. In this paper, we prove that, if G is a connected cactus and m(G)⩾7m(G)⩾7, then the b-chromatic number of G is at least m(G)−1m(G)−1.
A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent... more A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. Given a graph G and integers k and ℓ, the Cocoloring Problem is the problem of deciding if G has a (k, ℓ)-cocoloring. It is known that determining the cochromatic number (the minimum k + ℓ such that G is (k, ℓ)-cocolorable) is NP-hard . In 2011, Bravo et al. obtained a polinomial time algorithm for P4sparse graphs . In this paper, we generalize this result by obtaining a polynomial time algorithm for (q, q − 4)-graphs for every fixed q, which are the graphs such that every subset of at most q vertices induces at most q − 4 induced P4's. P4-sparse graphs are (5, 1)-graphs. Moreover, we prove that the cocoloring problem is FPT when parameterized by the treewidth tw(G) or by the parameter q(G), defined as the minimum integer q ≥ 4 such that G is a (q, q − 4)-graph.
ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯&lt... more ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 <v 2 <⋯<v n of V(G) such that v i has at least one neighbour in {v 1 ,⋯,v i-1 }, for every i∈{2,⋯,n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G)=2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k )>χ(G k ), for every k≥3. We then prove that for every graph G,χ c (G)≤χ(G)+1. Finally, we prove that deciding if χ c (G)=χ(G), given a graph G, is a NP-hard problem.
The Grundy number of a graph G is the largest k such that G has a greedy k-colouring, that is, a ... more The Grundy number of a graph G is the largest k such that G has a greedy k-colouring, that is, a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we give new bounds on the Grundy number of the product of two graphs.
A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent... more A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k + ℓ and the minimum max{k, ℓ} such that G is (k, ℓ)-cocolorable, is NP-hard problem. A (q, q−4)-graph is a graph such that every subset of at most q vertices induces at most q−4 distinct P 4 's. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a (5, 1)-graph is (k, ℓ)-cocolorable . In this paper, we extend this result by obtaining polynomial time algorithms to decide the (k, ℓ)-cocolorability and to determine the cochromatic number and the split chromatic number for (q, q − 4)-graphs for every fixed q and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum (k, ℓ)-cocolorable subgraph of a (q, q − 4)-graph for every fixed q. All these algorithms are fixed parameter tractable.
q, q − 4)-graphs Primeval decomposition Fixed parameter tractable algorithms a b s t r a c t Give... more q, q − 4)-graphs Primeval decomposition Fixed parameter tractable algorithms a b s t r a c t Given a graph G, a greedy coloring of G is a proper coloring such that, for each two colors i < j, every vertex of G colored j has a neighbor colored i. The Grundy number is the maximum number of colors in a greedy coloring of G. proved that determining the Grundy number is NP-hard even for complements of bipartite graphs. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other color class. The b-chromatic number is the maximum number of colors in a b-coloring of G. proved that determining the b-chromatic number is NP-hard. In this paper, we obtain polynomial time algorithms to determine the Grundy number and the b-chromatic number of (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set of at most q vertices induces at most q − 4 distinct P 4 . These algorithms are fixed parameter tractable on the parameter q(G), where q(G) is the minimum q such that G is a (q, q − 4)-graph.
The representatives formulation for the vertex coloring problem is revisited to remove symmetry a... more The representatives formulation for the vertex coloring problem is revisited to remove symmetry and new versions of facets derived from substructures of the graph are presented. In addition, a new class of facets is derived from independent sets of the graph. Finally, a comparison with the independent sets formulation is provided.
In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, th... more In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P 4 -tidy graphs and (q, q − 4)-graphs, for every fixed q. These classes include cographs, P 4 -sparse and P 4 -lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of (q, q − 4)-graphs. All these coloring problems are known to be NP-hard for general graphs.
The representatives formulation for the vertex coloring problem is revisited to remove symmetry a... more The representatives formulation for the vertex coloring problem is revisited to remove symmetry and new versions of facets derived from substructures of the graph are presented. In addition, a new class of facets is derived from independent sets of the graph. Finally, a comparison with the independent sets formulation is provided.
The Seventh European Conference on Combinatorics, Graph Theory and Applications, 2013
A b-coloring of a graph is a proper coloring of its vertices such that every color class contains... more A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b(G) such that the graph has a b-coloring with b(G) colors. This metric is upper bounded by the largest integer m(G) for which G has at least m(G) vertices with degree at least m(G)−1.
A b-coloring of a graph is a coloring of its vertices such that every color class contains a vert... more A b-coloring of a graph is a coloring of its vertices such that every color class contains a vertex that has a neighbor in all other classes. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. We show how to compute in polynomial time the b-chromatic number of a graph of girth at least 9. This improves the seminal result of Irving and Manlove on trees. *
O número cromático fracionário ( ) F G χ de um grafo G é um conhecido limite inferior para seu nú... more O número cromático fracionário ( ) F G χ de um grafo G é um conhecido limite inferior para seu número cromático ( ) G χ . Experimentos relatados na literatura mostram que usar ( ) F G χ , em lugar do tamanho da clique máxima, pode ser muito mais eficiente para orientar a busca em um algoritmo tipo branch-and-bound para determinação de ( ) G χ . Uma dificuldade, porém, é tratar o modelo linear conhecido para ( ) F G χ , o qual apresenta um número exponencial de variáveis e demanda um caro processo de geração de colunas. Neste trabalho, examinamos uma formulação alternativa para obter um limite inferior para ( ) F G χ que possui um número de variáveis linear no tamanho do grafo, porém um número exponencial de restrições. Utilizamos o método de planos-de-corte para lidar com esse inconveniente. Algumas heurísticas de separação são propostas, e experimentos computacionais mostram que valores muito próximos de ( ) F G χ , em muitos casos iguais, são encontrados em tempo inferior à implementação com geração de colunas.
A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex t... more A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G)χb(G) is the maximum number of colours used in a b-colouring of G . If m(G)m(G) is the largest integer k such that G has at least k vertices with degree at least k−1k−1, then we know that χb(G)⩽m(G)χb(G)⩽m(G). Irving and Manlove [Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, 91 (1999), pages 127–141] prove that, if T is a tree, then the b-chromatic number of T is at least m(T)−1m(T)−1. In this paper, we prove that, if G is a connected cactus and m(G)⩾7m(G)⩾7, then the b-chromatic number of G is at least m(G)−1m(G)−1.
A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent... more A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. Given a graph G and integers k and ℓ, the Cocoloring Problem is the problem of deciding if G has a (k, ℓ)-cocoloring. It is known that determining the cochromatic number (the minimum k + ℓ such that G is (k, ℓ)-cocolorable) is NP-hard . In 2011, Bravo et al. obtained a polinomial time algorithm for P4sparse graphs . In this paper, we generalize this result by obtaining a polynomial time algorithm for (q, q − 4)-graphs for every fixed q, which are the graphs such that every subset of at most q vertices induces at most q − 4 induced P4's. P4-sparse graphs are (5, 1)-graphs. Moreover, we prove that the cocoloring problem is FPT when parameterized by the treewidth tw(G) or by the parameter q(G), defined as the minimum integer q ≥ 4 such that G is a (q, q − 4)-graph.
ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 &lt;v 2 &lt;⋯&lt... more ABSTRACT A connected vertex ordering of a graph G is an ordering v 1 &lt;v 2 &lt;⋯&lt;v n of V(G) such that v i has at least one neighbour in {v 1 ,⋯,v i-1 }, for every i∈{2,⋯,n}. A connected greedy colouring is a colouring obtained by the greedy algorithm applied to a connected vertex ordering. In this paper we study the parameter Γ c (G), which is the maximum k such that G admits a connected greedy k-colouring, and χ c (G), which is the minimum k such that a connected greedy k-colouring of G exists. We prove that computing Γ c (G) is NP-hard for chordal graphs and complements of bipartite graphs. We also prove that if G is bipartite, Γ c (G)=2. Concerning χ c (G), we first show that there is a k-chromatic graph G k with χ c (G k )&gt;χ(G k ), for every k≥3. We then prove that for every graph G,χ c (G)≤χ(G)+1. Finally, we prove that deciding if χ c (G)=χ(G), given a graph G, is a NP-hard problem.
The Grundy number of a graph G is the largest k such that G has a greedy k-colouring, that is, a ... more The Grundy number of a graph G is the largest k such that G has a greedy k-colouring, that is, a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we give new bounds on the Grundy number of the product of two graphs.
A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent... more A (k, ℓ)-cocoloring of a graph G is a partition of the vertex set of G into at most k independent sets and at most ℓ cliques. It is known that determining the cochromatic number and the split chromatic number, which are respectively the minimum k + ℓ and the minimum max{k, ℓ} such that G is (k, ℓ)-cocolorable, is NP-hard problem. A (q, q−4)-graph is a graph such that every subset of at most q vertices induces at most q−4 distinct P 4 's. In 2011, Bravo et al. obtained a polynomial time algorithm to decide if a (5, 1)-graph is (k, ℓ)-cocolorable . In this paper, we extend this result by obtaining polynomial time algorithms to decide the (k, ℓ)-cocolorability and to determine the cochromatic number and the split chromatic number for (q, q − 4)-graphs for every fixed q and for graphs with bounded treewidth. We also obtain a polynomial time algorithm to obtain the maximum (k, ℓ)-cocolorable subgraph of a (q, q − 4)-graph for every fixed q. All these algorithms are fixed parameter tractable.
q, q − 4)-graphs Primeval decomposition Fixed parameter tractable algorithms a b s t r a c t Give... more q, q − 4)-graphs Primeval decomposition Fixed parameter tractable algorithms a b s t r a c t Given a graph G, a greedy coloring of G is a proper coloring such that, for each two colors i < j, every vertex of G colored j has a neighbor colored i. The Grundy number is the maximum number of colors in a greedy coloring of G. proved that determining the Grundy number is NP-hard even for complements of bipartite graphs. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other color class. The b-chromatic number is the maximum number of colors in a b-coloring of G. proved that determining the b-chromatic number is NP-hard. In this paper, we obtain polynomial time algorithms to determine the Grundy number and the b-chromatic number of (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set of at most q vertices induces at most q − 4 distinct P 4 . These algorithms are fixed parameter tractable on the parameter q(G), where q(G) is the minimum q such that G is a (q, q − 4)-graph.
The representatives formulation for the vertex coloring problem is revisited to remove symmetry a... more The representatives formulation for the vertex coloring problem is revisited to remove symmetry and new versions of facets derived from substructures of the graph are presented. In addition, a new class of facets is derived from independent sets of the graph. Finally, a comparison with the independent sets formulation is provided.
In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, th... more In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number and the harmonious chromatic number of P 4 -tidy graphs and (q, q − 4)-graphs, for every fixed q. These classes include cographs, P 4 -sparse and P 4 -lite graphs. We also obtain a polynomial time algorithm to determine the Grundy number of (q, q − 4)-graphs. All these coloring problems are known to be NP-hard for general graphs.
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