A Spectral Technique for Coloring Random 3-Colorable Graphs

18 until the number of vertices left is O(n=d), and color the remaining vertices in an arbitrary manner. 7. The existence of an approximation algorithm based on the spectral method for coloring arbitrary graphs is a question that deserves further investigation (which we do not address here.) Recently, improved approximation algorithms for graph coloring have been obtained using semideenite programming 14], 5]. Acknowledgement We thank two anonymous referees for several suggestions that improved the presentation of the paper. 17 based on eigenvectors of the adjacency matrix, and showed that their heuristic optimally colors complete 3-partite graphs as well as certain other classes of graphs with regular structure. 2. By modifying some of the arguments of Section 2 we can show that if p is somewhat bigger (p log 3 n=n suuces) then almost surely the initial coloring V 0 i that is computed from the eigenvectors e 3n?1 and e 3n in the rst phase of our algorithm is completely correct. In this case the last two phases of the algorithm are not needed. By reening the argument in Subsection 2.2, it can also be shown that if p > 10 log n=n the third phase of the algorithm is not needed, and the coloring obtained by the end of the second phase will almost surely be the correct one. 3. We can show that a variant of our algorithm nds, almost surely, a proper coloring in the model of random regular 3-colorable graphs in which one chooses randomly d perfect match-ings between each pair of distinct color classes, when d is a suuciently large absolute constant. Here, in fact, the proof is simpler, as the smallest two eigenvalues (and their corresponding eigenspace) are known precisely, as noted in Subsection 1.2. 4. The results easily extend to the model in which each vertex rst picks a color randomly, independently and uniformly, among the three possibilities, and next every pair of vertices of distinct colors becomes an edge with probability p (> c=n). 5. If G = G 3n;p;3 and p c=n for some small positive constant c, it is not diicult to show that almost surely G does not have any subgraph with minimum degree at least 3, and hence it is easy to 3-color it by a greedy-type (linear time) algorithm. For values of p which are bigger than this c=n but satisfy p = o(log n=n), the graph …