The b-chromatic number of power graphs

The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles

On The b-Chromatic Number of Regular Graphs Without 4-Cycle

The b-chromatic number of a graph $G$, denoted by $\phi(G)$, is the largest integer $k$ that $G$ admits a proper $k$-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 $d$-regular graph $G$ which contains no 4-cycle, $\phi(G)\geq\lfloor\frac{d+3}{2}\rfloor$ and if $G$ has a triangle, then $\phi(G)\geq\lfloor\frac{d+4}{2}\rfloor$. Also, if $G$ is a $d$-regular graph which contains no 4-cycle and $diam(G)\geq6$, then $\phi(G)=d+1$. Finally, we show that for any $d$-regular graph $G$ which does not contain 4-cycle and $\kappa(G)\leq\frac{d+1}{2}$, $\phi(G)=d+1$

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

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

Contributions au tri automatique de documents et de courrier d'entreprises

Ce travail de thèse s inscrit dans le cadre du développement de systèmes de vision industrielle pour le tri automatique de documents et de courriers d entreprises. Les architectures existantes, dont nous avons balayé les spécificités dans les trois premiers chapitres de la thèse, présentent des faiblesses qui se traduisent par des erreurs de lecture et des rejets que l on impute encore trop souvent aux OCR. Or, les étapes responsables de ces rejets et de ces erreurs de lecture sont les premières à intervenir dans le processus. Nous avons ainsi choisi de porter notre contribution sur les aspects inhérents à la segmentation des images de courriers et la localisation de leurs régions d intérêt en investissant une nouvelle approche pyramidale de modélisation par coloration hiérarchique de graphes ; à ce jour, la coloration de graphes n a jamais été exploitée dans un tel contexte. Elle intervient dans notre contribution à toutes les étapes d analyse de la structure des documents ainsi que dans la prise de décision pour la reconnaissance (reconnaissance de la nature du document à traiter et reconnaissance du bloc adresse). Notre architecture a été conçue pour réaliser essentiellement les étapes d analyse de structures et de reconnaissance en garantissant une réelle coopération entres les différents modules d analyse et de décision. Elle s articule autour de trois grandes parties : une partie de segmentation bas niveau (binarisation et recherche de connexités), une partie d extraction de la structure physique par coloration hiérarchique de graphe et une partie de localisation de blocs adresse et de classification de documents. Les algorithmes impliqués dans le système ont été conçus pour leur rapidité d exécution (en adéquation avec les contraintes de temps réels), leur robustesse, et leur compatibilité. Les expérimentations réalisées dans ce contexte sont très encourageantes et offrent également de nouvelles perspectives à une plus grande diversité d images de documents.This thesis deals with the development of industrial vision systems for automatic business documents and mail sorting. These systems need very high processing time, accuracy and precision of results. The current systems are most of time made of sequential modules needing fast and efficient algorithms throughout the processing line: from low to high level stages of analysis and content recognition. The existing architectures that we have described in the three first chapters of the thesis have shown their weaknesses that are expressed by reading errors and OCR rejections. The modules that are responsible of these rejections and reading errors are mostly the first to occur in the processes of image segmentation and interest regions location. Indeed, theses two processes, involving each other, are fundamental for the system performances and the efficiency of the automatic sorting lines. In this thesis, we have chosen to focus on different sides of mail images segmentation and of relevant zones (as address block) location. We have chosen to develop a model based on a new pyramidal approach using a hierarchical graph coloring. As for now, graph coloring has never been exploited in such context. It has been introduced in our contribution at every stage of document layout analysis for the recognition and decision tasks (kind of document or address block recognition). The recognition stage is made about a training process with a unique model of graph b-coloring. Our architecture is basically designed to guarantee a good cooperation bewtween the different modules of decision and analysis for the layout analysis and the recognition stages. It is composed of three main sections: the low-level segmentation (binarisation and connected component labeling), the physical layout extraction by hierarchical graph coloring and the address block location and document sorting. The algorithms involved in the system have been designed for their execution speed (matching with real time constraints), their robustness, and their compatibility. The experimentations made in this context are very encouraging and lead to investigate a wider diversity of document images.VILLEURBANNE-DOC'INSA-Bib. elec. (692669901) / SudocSudocFranceF

The b-chromatic number of power graphs

International audienceThe b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles