Why Almost All k-Colorable Graphs Are Easy to Color

Coloring a k-colorable graph using k colors (k ≥ 3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs are clustered in one cluster, and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.

• Type: Journal
• Year: 2010
• Where: MST
• Authors: Amin Coja-Oghlan, Michael Krivelevich, Dan Vilenchik
• Keywords: K-colorable Graph, MST 2010, Proper K-coloring, Sparse Random Graphs