The b-chromatic number and related topics—A survey

Jakovac, Marko; Peterin, Iztok

doi:10.1016/j.dam.2017.08.008

Cited by 21 publications

(13 citation statements)

References 50 publications

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“…Later, Cabello and Jakovac (2011) proved that for every d-regular graph with at least 2d 3 vertices χ b (G) = ∆ + 1. A detailed review of the literature related to the b-chromatic number can be found in Jakovac and Peterin (2018).…”

Section: Introduction 1basic Notation and Problem Definitionmentioning

confidence: 99%

A matheuristic approach for the $b$-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic

Melo,

Queiroz,

Santos

2021

Preprint

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Given a graph G = (V, E), the b-coloring problem consists in attributing a color to every vertex in V such that adjacent vertices receive different colors, every color has a b-vertex, and the number of colors is maximized. A b-vertex is a vertex adjacent to vertices colored with all used colors but its own. The b-coloring problem is known to be NP-Hard and its optimal solution determines the b-chromatic number of G, denoted χ b (G). This paper presents an integer programming formulation and a very effective multi-greedy randomized heuristic which can be used in a multi-start metaheuristic. In addition, a matheuristic approach is proposed combining the multi-start multi-greedy randomized metaheuristic with a MIP (mixed integer programming) based local search procedure using the integer programming formulation. Computational experiments establish the proposed multi-start metaheuristic as very effective in generating high quality solutions, along with the matheuristic approach successfully improving several of those results. Moreover, the computational results show that the multi-start metaheuristic outperforms a state-of-the-art hybrid evolutionary metaheuristic for a subset of the large instances which were previously considered in the literature. An additional contribution of this work is the proposal of a benchmark instance set, which consists of newly generated instances as well as others available in the literature for classical graph problems, with the aim of standardizing computational comparisons of approaches for the b-coloring problem in future works.

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Section: Introduction 1basic Notation and Problem Definitionmentioning

confidence: 99%

A matheuristic approach for the $b$-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic

Melo,

Queiroz,

Santos

2021

Preprint

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“…Also by a color dominating vertex u of G we mean the vertex u has at least one neighbor in each color class of the color-dominating coloring of G. Denote by b(G) the maximum number of colors used in any color-dominating coloring of G. The b-chromatic number has been widely studied in graph theory [2,3,4,7,8,10,11,12,13]. For a survey on b-chromatic number, we refer to [9]. In this paper we call a graph G b-monotone if for each induced subgraph H of G we have b(H) ≤ b(G).…”

Section: Introductionmentioning

confidence: 99%

On Grundy and b-chromatic number of some families of graphs: a comparative study

Masih¹,

Zaker²

2020

Preprint

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The Grundy and the b-chromatic number of graphs are two important chromatic parameters. The Grundy number of a graph G, denoted by Γ(G) is the worst case behavior of greedy (First-Fit) coloring procedure for G and the b-chromatic number b(G) is the maximum number of colors used in any color-dominating coloring of G. Because the nature of these colorings are different they have been studied widely but separately in the literature. This paper presents a comparative study of these coloring parameters. There exists a sequence {G n } n≥1 with limited b-chromatic number but Γ(G n ) → ∞. We obtain families of graphs F such that for some adequate function f (.), Γ(G) ≤ f (b(G)), for each graph G from the family. This verifies a previous conjecture for these families.

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“…The b-chromatic number b(G) is the largest integer k such that G admits a b-coloring with k colors. Since this parameter has been introduced by R. W. Irving and D. F. Manlove [6], it aroused the interest of many researchers as we can see in [5], [2] and [3] and, more recently, in [7].…”

Section: Introductionmentioning

confidence: 99%

New bounds for the b-chromatic number of vertex deleted graphs

Del-Vecchio¹,

Kouider²

2019

Preprint

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A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs.We also get bounds for the b-chromatic number of G − {x}, when G is a graph with large girth.

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The b-chromatic number and related topics—A survey

Cited by 21 publications

References 50 publications

A matheuristic approach for the $b$-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic

A matheuristic approach for the $b$-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic

On Grundy and b-chromatic number of some families of graphs: a comparative study

New bounds for the b-chromatic number of vertex deleted graphs

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