Bert Randerath

Bio: Bert Randerath is an academic researcher from University of Cologne. The author has contributed to research in topics: Chordal graph & 1-planar graph. The author has an hindex of 16, co-authored 39 publications receiving 782 citations. Previous affiliations of Bert Randerath include Cologne University of Applied Sciences & RWTH Aachen University.

Vertex Colouring and Forbidden Subgraphs – A Survey

Abstract: There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).

3-Colorability ∈ P for P 6 -free graphs

Abstract: In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.

Polynomial $$\chi $$ χ -Binding Functions and Forbidden Induced Subgraphs: A Survey

Abstract: A graph G with clique number $$\omega (G)$$ and chromatic number $$\chi (G)$$ is perfect if $$\chi (H)=\omega (H)$$ for every induced subgraph H of G. A family $${\mathcal {G}}$$ of graphs is called $$\chi $$ -bounded with binding function f if $$\chi (G') \le f(\omega (G'))$$ holds whenever $$G \in {\mathcal {G}}$$ and $$G'$$ is an induced subgraph of G. In this paper we will present a survey on polynomial $$\chi $$ -binding functions. Especially we will address perfect graphs, hereditary graphs satisfying the Vizing bound ( $$\chi \le \omega +1$$ ), graphs having linear $$\chi $$ -binding functions and graphs having non-linear polynomial $$\chi $$ -binding functions. Thereby we also survey polynomial $$\chi $$ -binding functions for several graph classes defined in terms of forbidden induced subgraphs, among them $$2K_2$$ -free graphs, $$P_k$$ -free graphs, claw-free graphs, and $${ diamond}$$ -free graphs. ( [])

A Theory of Graphs

Abstract: A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

Discrete Applied Mathematics

Abstract: : Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

Advances on the Hamiltonian Problem – A Survey

Abstract: This article is intended as a survey, updating earlier surveys in the area. For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs and digraphs.