TY - JOUR

T1 - A spectral technique for coloring random 3-colorable graphs

AU - Alon, Noga

AU - Kahale, Nabil

PY - 1997/12

Y1 - 1997/12

N2 - Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p ≥ c/n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p ≥ polylog(n)/n [J. Algorithms, 19 (1995), pp. 204-234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c.

AB - Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p ≥ c/n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p ≥ polylog(n)/n [J. Algorithms, 19 (1995), pp. 204-234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c.

KW - Algorithms

KW - Graph coloring

KW - Graph eigenvalues

KW - Random graphs

UR - http://www.scopus.com/inward/record.url?scp=0001690318&partnerID=8YFLogxK

U2 - 10.1137/S0097539794270248

DO - 10.1137/S0097539794270248

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AN - SCOPUS:0001690318

SN - 0097-5397

VL - 26

SP - 1733

EP - 1748

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 6

ER -