13 research outputs found
On b-chromatic Number of Prism Graph Families
A b-coloring of graph is a proper -coloring that verifies the following property: for every color class , 1≤≤, there exists a vertex , with color , such that all the other colors in are utilized in neighbors. The b-chromatic number of a graph , denoted by (), is the largest integer such that may have a b-coloring by colors. In this paper we discuss the b-coloring of prism graph , central graph of prism graph (), middle graph of prism graph () and the total graph of prism graph () and we obtain the b-chromatic number for these graphs
On The b-Chromatic Number of Regular Graphs Without 4-Cycle
The b-chromatic number of a graph , denoted by , is the largest
integer that admits a proper -coloring such that each color class
has a vertex that is adjacent to at least one vertex in each of the other color
classes. We prove that for each -regular graph which contains no
4-cycle, and if has a triangle,
then . Also, if is a -regular
graph which contains no 4-cycle and , then .
Finally, we show that for any -regular graph which does not contain
4-cycle and ,
The b-Chromatic Number of Star Graph Families
In this paper, we investigate the b-chromatic number of central graph, middle graph and total graph of star graph, denoted by C(K1,n), M(K1,n) and T(K1,n) respectively. We discuss the relationship between b-chromatic number with some other types of chromatic numbers such as chromatic number, star chromatic number and equitable chromatic number
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
The b-Chromatic Number of Regular Graphs via The Edge Connectivity
\noindent The b-chromatic number of a graph , denoted by , is the
largest integer that admits a proper coloring by colors, such that
each color class has a vertex that is adjacent to at least one vertex in each
of the other color classes. El Sahili and Kouider [About b-colorings of regular
graphs, Res. Rep. 1432, LRI, Univ. Orsay, France, 2006] asked whether it is
true that every -regular graph of girth at least 5 satisfies
. Blidia, Maffray, and Zemir [On b-colorings in regular graphs,
Discrete Appl. Math. 157 (2009), 1787-1793] showed that the Petersen graph
provides a negative answer to this question, and then conjectured that the
Petersen graph is the only exception. In this paper, we investigate a
strengthened form of the question.
The edge connectivity of a graph , denoted by , is the minimum
cardinality of a subset of such that is either
disconnected or a graph with only one vertex. A -regular graph is called
super-edge-connected if every minimum edge-cut is the set of all edges incident
with a vertex in , i.e., and every minimum edge-cut of
isolates a vertex. We show that if is a -regular graph that contains no
4-cycle, then whenever is not super-edge-connected
A new approach to b-coloring of regular graphs
Let be a graph and c a proper k-coloring of G, i.e. any two adjacent
vertices u and v have different colors c(u) and c(v). A proper k-coloring is a
b-coloring if there exists a vertex in every color class that contains all the
colors in its closed neighborhood. The maximum number of colors k admitting
b-coloring of G is the b-chromatic number. We present two separate approaches
to the conjecture posed by Blidia et. al that the b-chromatic number equals to
d+1 for every d-regular graph of girth at least five except the Petersen graph.Comment: 16 pages, 2 figures, 13 reference
On b-colorings and b-continuity of graphs
A b-coloring of G is a proper vertex coloring such that there is a vertex in each color class, which is adjacent to at least one vertex in every other color class. Such a vertex is called a color-dominating vertex. The b-chromatic number of G is the largest k such that there is a b-coloring of G by k colors.
Moreover, if for every integer k, between chromatic number and b-chromatic number, there exists a b-coloring of G by k colors, then G is b-continuous. Determining the b-chromatic number of a graph G and the decision whether the given graph G is b-continuous or not is NP-hard. Therefore, it is interesting to find new results on b-colorings and b-continuity for special graphs.
In this thesis, for several graph classes some exact values as well as bounds of the b-chromatic number were ascertained. Among all we considered graphs whose independence number, clique number, or minimum degree is close to its order as well as bipartite graphs. The investigation of bipartite graphs was based on considering of the so-called bicomplement which is used to determine the b-chromatic number of special bipartite graphs, in particular those whose bicomplement has a simple structure. Then we studied some graphs whose b-chromatic number is close to its t-degree.
At last, the b-continuity of some graphs is studied, for example, for graphs whose b-chromatic number was already established in this thesis. In particular, we could prove that Halin graphs are b-continuous.:Contents
1 Introduction
2 Preliminaries
2.1 Basic terminology
2.2 Colorings of graphs
2.2.1 Vertex colorings
2.2.2 a-colorings
3 b-colorings
3.1 General bounds on the b-chromatic number
3.2 Exact values of the b-chromatic number for special graphs
3.2.1 Graphs with maximum degree at most 2
3.2.2 Graphs with independence number close to its order
3.2.3 Graphs with minimum degree close to its order
3.2.4 Graphs G with independence number plus clique number at most number of vertices
3.2.5 Further known results for special graphs
3.3 Bipartite graphs
3.3.1 General bounds on the b-chromatic number for bipartite graphs
3.3.2 The bicomplement
3.3.3 Bicomplements with simple structure
3.4 Graphs with b-chromatic number close to its t-degree
3.4.1 Regular graphs
3.4.2 Trees and Cacti
3.4.3 Halin graphs
4 b-continuity
4.1 b-spectrum of special graphs
4.2 b-continuous graph classes
4.2.1 Known b-continuous graph classes
4.2.2 Halin graphs
4.3 Further graph properties concerning b-colorings
4.3.1 b-monotonicity
4.3.2 b-perfectness
5 Conclusion
Bibliograph