Coloring Sparse Random Graphs in Polynomial Average Time

Abstract

We present a simple BFS tree approach to solve partitioning problems on random graphs. In this paper, we use this approach to study the k-coloring problem. Consider random k-colorable graphs drawn by choosing each allowed edge independently with probability p after arbitrarily partitioning the vertex set into k color classes of “roughly equal” (i.e. Ω(n)) sizes. Given a graph G and two vertices x and y, compute n(G,x,y) as follows: Grow the BFS tree (in the subgraph induced by V - y) from x till the l-th level (for some suitable l) and find the number of neighbors of y in this level. We show that these quantities computed for all pairs are sufficient to separate the largest or smallest color class. Repeating this procedure k - 1 times, one obtains a k-coloring of G with high probability, if p ≥ n^{-1 + ∈}, ∈ ≥ X/$\sqrt {\log {\mathbf{ }}n}$ for some large constant X. We also show how to use this approach so that one gets even smaller failure probability at the cost of running time. Based on this, we present polynomial average time (p.av.t. ) k-coloring algorithms for the stated range of p. This improves significantly previous results on p.av.t. coloring [13] where a is required to be above 1/4. Previous works on coloring random graphs have been mostly concerned with almost surely succeeding (a.s. ) algorithms and little work has been done on p.av.t. algorithms. An advantage of the BFS approach is that it is conceptually very simple and combinatorial in nature. This approach is applicable to other partitioning problems also.

This research was partially supported by the EU ESPRIT LTR Project No. 20244 (ALCOM-IT), WP 3.3 and also by a fellowship of MPI(Informatik), Saarbrücken, Germany.