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

b-chromatic number of cacti

A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G) is the maximum number of colours used in a b-colouring of G. If m(G) is the largest integer k such that G has at least k vertices with degree at least k−1, then we know that χb(G)m(G). Irving and Manlove [Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, 91 (1999), pages 127-141] prove that, if T is a tree, then the b-chromatic number of T is at least m(T)−1. In this paper, we prove that, if G is a connected cactus and m(G)7, then the b-chromatic number of G is at least m(G)−1