A randomised 3-colouring algorithm
Discrete Mathematics - Graph colouring and variations
Sudden emergence of a giant k-core in a random graph
Journal of Combinatorial Theory Series B
A sharp threshold for k-colorability
Random Structures & Algorithms
Almost all graphs with average degree 4 are 3-colorable
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Analysis of Edge Deletion Processes on Faulty Random Regular Graphs
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
The analysis of a list-coloring algorithm on a random graph
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth
Combinatorics, Probability and Computing
The pure literal rule threshold and cores in random hypergraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
| Hi-index | 0.00 |
We describe a novel subgraph of k-partite graphs suddenly appearing at an average degree c = 4.91 ... (for k = 3) in random graphs with a built-in k-partition. These magic subgraphs consist of directed edges and comprise a constant fraction of the nodes, as soon as they appear. The phenomenon is similar to the Sudden Emergence of a Giant k-Core [B. Pittel, J. Spence, N. Wormald, Sudden emergence of a giant k-core in a random graph, J. Combin. Theory Ser. B 67 (1996) 111-151] and can be easily demonstrated in simulations. Thus generated magic subgraphs appear to be 'almost' uniquely colourable. On the theoretical side, we give an indication how central parts of our novel proof for the aforementioned k-core phenomenon [U. Voll, Threshold phenomena in branching trees and random graphs, Ph.D. Thesis, Lehrstuhl für Effiziente Algorithmen, Technische Universität München, Germany, 2001.] can be modified in order to prove the sudden appearance of a subgraph which is (obviously) closely related to the empirically observed magic subgraph, appearing at the right critical average degree and having the right size compared to simulations. We conclude with discussing a number of open questions related to the magic subgraph.