b-Coloring graphs with large girth

Campos, Víctor; Farias, Victor A. E. de; Silva, Ana

doi:10.1007/s13173-012-0063-9

Cited by 8 publications

(3 citation statements)

References 20 publications

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“…In contrast, the exact result for ϕ(T ) for every tree T was presented already in Irving and Manlove (1999). A similar approach was later transformed to cactus graphs (Campos et al 2009), to outerplanar graphs (Maffray and Silva 2012), and to graphs with large enough girth (Campos et al 2012(Campos et al , 2015Kouider and Zamime 2017). For further reading about b-chromatic number and related concepts we recommend survey (Jakovac and Peterin 2018).…”

Section: Introductionmentioning

confidence: 99%

On b-acyclic chromatic number of a graph

2022

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Let G be a graph. We introduce the acyclic b-chromatic number of G as an analogue to the b-chromatic number of G. An acyclic coloring of a graph G is a map $$c:V(G)\rightarrow \{1,\ldots ,k\}$$ c : V ( G ) → { 1 , … , k } such that $$c(u)\ne c(v)$$ c ( u ) ≠ c ( v ) for any $$uv\in E(G)$$ u v ∈ E ( G ) and the induced subgraph on vertices of any two colors $$i,j\in \{1,\ldots ,k\}$$ i , j ∈ { 1 , … , k } induces a forest. On the set of all acyclic colorings of G we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number A(G) of G and the maximum cardinality of its minimal element is the acyclic b-chromatic number $$A_{\textrm{b}}(G)$$ A b ( G ) of G. We present several properties of $$A_{\textrm{b}}(G).$$ A b ( G ) . In particular, we derive $$A_{\textrm{b}}(G)$$ A b ( G ) for several known graph families, derive some bounds for $$A_{\textrm{b}}(G),$$ A b ( G ) , compare $$A_{\textrm{b}}(G)$$ A b ( G ) with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.

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Section: Introductionmentioning

confidence: 99%

On b-acyclic chromatic number of a graph

2022

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“…However, this problem is polynomial when restricted to some graph classes, including trees [12], cographs and P 4 -sparse graphs [3], P 4 -tidy graphs [20], cacti [18], some power graphs [5][6][7], Kneser graphs [9,14], ✩ Partially supported by CAPES, FUNCAP and CNPq/Brazil. some graphs with large girth [8,15,17], etc. Also, some other aspects of the problem were studied, as for example, the bspectrum of a graph [1], and b-perfect graphs [11].…”

Section: Introductionmentioning

confidence: 99%

The b-chromatic index of graphs

Campos

Lima

Martins

et al. 2015

Discrete Mathematics

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a b s t r a c tA b-coloring of the vertices of a graph is a proper coloring where each color class contains a vertex which is adjacent to at least one vertex in each other color class. The b-chromatic number of G is the maximum integer b(G) for which G has a b-coloring with b(G) colors. This problem was introduced by Irving and Manlove (1999), where they showed that computing b(G) is N P -hard in general and polynomial-time solvable for trees. A natural question that arises is whether the edge version of this problem is also N P -hard or not. Here, we prove that computing the b-chromatic index of a graph G is N P -hard, even if G is either a comparability graph or a C k -free graph, and give partial results on the complexity of the problem restricted to trees, more specifically, we solve the problem for caterpillars graphs. Although solving problems on caterpillar graphs is usually quite simple, this problem revealed itself to be unusually hard. The presented algorithm uses a dynamic programming approach that combines partial solutions which are proved to exist if, and only if, a particular polyhedron is non-empty.

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“…Analysis of the b-chromatic number for special graph classes such as powers of paths, cycles, complete binary trees, and complete caterpillars can be found in [10][11][12]. Further graph classes, such as cacti [7], Kneser graphs [21], cographs, P 4 sparse [4], P 4 tidy graphs [30] and large girth graphs [28,6], were more recently considered too. Bounds for the b-chromatic number were studied in turn in [2,1,8,23,25].…”

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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b-Coloring graphs with large girth

Cited by 8 publications

References 20 publications

On b-acyclic chromatic number of a graph

On b-acyclic chromatic number of a graph

The b-chromatic index of graphs

The b-chromatic index of direct product of graphs

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