Alapfogalmak
No perfect graph is ugly, and good connected orderings can be computed efficiently.
Kivonat
The article discusses connected greedy colourings of perfect graphs and other classes. It explores the concepts of Grundy numbers, greedy colouring algorithms, and different subclasses of graphs. The main focus is on proving that no perfect graph is ugly by providing constructive proofs for various subclasses of perfect graphs. The content is structured into sections covering different aspects of greedy colouring algorithms, including K4-minor-free graphs, comparability graphs, and Meyniel graphs. The authors present detailed proofs and algorithms to support their claims.
Structure:
Introduction to Greedy Colouring Algorithms
Overview of the complexity of graph colouring problems.
Grundy Numbers and Greedy Colouring
Explanation of Grundy numbers in relation to greedy colouring.
Connected Orderings and Graph Classes
Discussion on connected orderings in different graph classes.
Results for Specific Graph Classes
Detailed proofs for K4-minor-free graphs, comparability graphs, and perfect graphs.
Algorithmic Approaches
Description of algorithms for computing good connected orderings efficiently.
Acknowledgements
Statisztikák
The Grundy number of a graph is the maximum number of colours used by the “First-Fit” greedy colouring algorithm over all vertex orderings.
No perfect graph is ugly.
For any positive integer k, any connected k-chromatic perfect graph G has a good connected ordering starting with any vertex v.
Idézetek
"No perfect graph is ugly."
"A good connected ordering can be computed in time O(nc+4)."