Some tools for approximate 3-coloring

Several tools for use in approximation algorithms to color 3-chromatic graphs are presented. The techniques are used in an algorithm that colors any 3-chromatic graph with O(n ^3/8)+O(n^3/8+O(1)) colors (or more precisely) O(n^3/8log^5/8 n) colors, which improves the previous best bound of O(n^0.4+0(1)) colors. The techniques are illustrated by considering a problem in which the 3-chromatic graph is created not by a worst-case adversary, but by an adversary each of whose decisions (whether or not to include an edge) is reversed with some small probability or noise rate p. This type of adversary is equivalent to the semirandom source of M. Santha and U.V. Vazirani (1986). An algorithm that will actually 3-color such a graph with high probability even for quite low noise rates (p⩾n ^-1/2+ε for constant ε>0), is presented