# Minimum Coloring Random and Semi-Random Graphs in Polynomial Expected Time

Subramanian, CR (1995) Minimum Coloring Random and Semi-Random Graphs in Polynomial Expected Time. In: 36th Annual Symposium on Foundations of Computer Science, 1995. Proceedings, 23-25 October 1995, Milwaukee, WI, USA, pp. 463-472.

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## Abstract

We present new algorithms for k-coloring and minimum (x(G)-) coloring random and semi-random k- colorable graphs in polynomial expected time. The random graphs are drawn from the G(n,p, k) model and the semi-random graphs are drawn from the $G_{SB}(n,p,k)$ model. In both models, an adversary initially splits the n vertices into K color classes, each of size $\Theta(n)$. Then the edges between vertices in different color classes are chosen one by one, according to some probability distribution. The model $G_{SB}(n,p,k)$ was introduced by Blum [3] and with respect to randomness, it lies between the random model G(n,p,k) where all edges are chosen with equal probability and the worst-case model. This extended abstract consists of two parts. In Part I, we propose a general methodology for designing algorithms for k-coloring random graphs from G(n,p,k). Using this, we derive new algorithms for k-coloring $G \in G(n,p,k)$ for $p\geq n^{-1+\epsilon}$ where E is any constant greater than 1/4. Our algorithms run in polynomial time on the average. This improves the results of [13] where \epsilon was required to be above 0.4. In Part II, we present polynomial average time algorithms for minimum coloring semi-random graphs from $G_{SB}(n,p,k)$ for $p \geq n^{-\alpha(k)+\epsilon}$, where \alpha(k) = (2IC)/((k - l)(k + 2)) and E is any positive constant. We also present polynomial average time algorithms for X(G)-coloring random graphs from G(n,p,k), with $p \geq n^{-\gamma(k)+\epsilon}$ where $\gamma(k) = (2k)/(k^2 - k + 2)$. The problem of X(G)-coloring is harder than the k-coloring problem because every k-colorable graph has a ”short certificate” for k-colorability, but there are many k-colorable graphs with x(G) = k for which there is no known short certificate for the fact x(G) = k.

Item Type: Conference Paper Copyright 1995 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 18 Sep 2007 11 Jan 2012 10:04 http://eprints.iisc.ernet.in/id/eprint/10961