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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:105-qucosa-91857 |
| Date | 24 July 2012 |
| Creators | Alkhateeb, Mais |
| Contributors | Technische Universität Bergakademie Freiberg, Mathematik und Infomatik, Prof. Dr. rer. nat. Ingo Schiermeyer, Prof. Dr. rer. nat. Ingo Schiermeyer, Prof. Dr. rer. nat. Margit Voigt |
| Publisher | Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola" |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf, application/pdf, application/pdf, application/pdf, application/zip |
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