# Coloring random graphs

## Abstract

We present an algorithm for coloring random 3-chromatic graphs with edge probabilities below the *n*^{−1/2} “barrier”. Our (deterministic) algorithm succeeds with high probability to 3-color a random 3-chromatic graph produced by partitioning the vertex set into three almost equal sets and selecting an edge between two vertices of different sets with probability *p*≥*n*^{−} 3/5+*ε*. The method is extended to *k*-chromatic graphs, succeeding with high probability for *p*≥*n*^{−α+ε} with α=2*k*/((*k*−l)(*k*+2)) and ε>0. The algorithms work also for Blum's balanced semi-random G_{SB}(*n,p,k*) model where an adversary chooses the edge probability up to a small additive noise *p*. In particular, our algorithm does not rely on any uniformity in the degree.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Blum, Some Tools for Approximate 3-Coloring, FOCS (1990), 554–562.Google Scholar
- 2.M.E. Dyer and A.M. Frieze, The Solution of Some Random NP-Hard Problems in Polynomial Expected Time,
*J. Alg.***10**(1989), 451–489.Google Scholar - 3.M. Santha and U.V. Vazirani, Generating Quasi-random Sequences from Semi-random Sources,
*J. Comp. Syst. Sci.***33**(1986), 75–87.Google Scholar - 4.J.S. Turner, Almost All
*k*-colorable Graphs are Easy to Color,*J. Alg.***9**(1988), 63–82.Google Scholar