The b-Chromatic Number of Cubic Graphs

Jakovac, Marko; Klavžar, Sandi

doi:10.1007/s00373-010-0898-9

Cited by 38 publications

(16 citation statements)

References 13 publications

(2 reference statements)

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“…The following theorem is proven by Jakovac and Klavzar in [15]. They showed that, except for four simple graphs, the b-chromatic number of connected cubic graph is 4.…”

Section: Some Known Resultsmentioning

confidence: 95%

On b-vertex and b-edge critical graphs

Eschouf¹,

Blidia²

2015

OpMath

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Abstract. A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the b-chromatic number b(G) of a graph G is the largest integer k such that G admits a b-coloring with k colors. A simple graph G is called b + -vertex (edge) critical if the removal of any vertex (edge) of G increases its b-chromatic number. In this note, we explain some properties in b + -vertex (edge) critical graphs, and we conclude with two open problems.

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“…The following theorem is proven by Jakovac and Klavzar in [15]. They showed that, except for four simple graphs, the b-chromatic number of connected cubic graph is 4.…”

Section: Some Known Resultsmentioning

confidence: 95%

On b-vertex and b-edge critical graphs

Eschouf¹,

Blidia²

2015

OpMath

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“…[8,19]). The b-chromatic number of regular graphs has been investigated in a series of papers [6,17,20,22]. Determining the b-chromatic number of a tight graph is NP-hard even for a connected bipartite graph [18] and a tight chordal graph [12].…”

Section: Tablementioning

confidence: 99%

A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

Effantin

Gastineau

Togni

2016

Discrete Mathematics

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International audienceGyarfas et al. and Zaker have proven that the Grundy number of a graph G satisfies Gamma(G) >= t if and only if G contains an induced subgraph called a t-atom. The family of t-atoms has bounded order and contains a finite number of graphs. In this article, we introduce equivalents of t-atoms for b-coloring and partial Grundy coloring. This concept is used to prove that determining if phi(G) >= t and partial derivative Gamma(G) >= t (under conditions for the b-coloring), for a graph G, is in XP with parameter t. We illustrate the utility of the concept of t-atoms by giving results on b-critical vertices and edges, on b-perfect graphs and on graphs of girth at least 7. (C) 2016 Elsevier B.V. All rights reserved

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“…This bound was later improved to 2d 3 in [5]. In particular, it was shown in [18] that there are only four exceptions among cubic graphs with ϕ(G) < 4, one of them being the Petersen graph.…”

Section: Introductionmentioning

confidence: 99%

The b-chromatic index of direct product of graphs

Koch

Peterin

2015

Discrete Applied Mathematics

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a b s t r a c tThe b-chromatic index ϕ ′ (G) of a graph G is the largest integer k such that G admits a proper k-edge coloring in which every color class contains at least one edge incident to edges in every other color class. We give in this work bounds for the b-chromatic index of the direct product of graphs and provide general results for many direct products of regular graphs. In addition, we introduce an integer linear programming model for the b-edge coloring problem, which we use for computing exact results for the direct product of some special graph classes.

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The b-Chromatic Number of Cubic Graphs

Cited by 38 publications

References 13 publications

On b-vertex and b-edge critical graphs

On b-vertex and b-edge critical graphs

A characterization of b-chromatic and partial Grundy numbers by induced subgraphs

The b-chromatic index of direct product of graphs

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