TY - JOUR

T1 - Why almost all k-colorable graphs are easy to color

AU - Coja-Oghlan, Amin

AU - Krivelevich, Michael

AU - Vilenchik, Dan

N1 - Funding Information:
Research of M. Krivelevich was supported in part by USA-Israel BSF Grant 2002-133, and by grant 526/05 from the Israel Science Foundation.

PY - 2010/4/1

Y1 - 2010/4/1

N2 - 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 lie in a single "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.

AB - 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 lie in a single "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.

KW - Average case analysis

KW - Random graphs

KW - Spectral analysis

KW - k-colorability

UR - http://www.scopus.com/inward/record.url?scp=77951258141&partnerID=8YFLogxK

U2 - 10.1007/s00224-009-9231-5

DO - 10.1007/s00224-009-9231-5

M3 - Article

AN - SCOPUS:77951258141

VL - 46

SP - 523

EP - 565

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 3

ER -