Weak coloring

A graph vertex coloring is a weak coloring, but not necessarily vice versa. Every graph has a weak 2-coloring. The figure on the right illustrates a simple algorithm for constructing a weak 2-coloring in an arbitrary graph. Part (a) shows the original graph. Part (b) shows a breadth-first search tree of the same graph. Part (c) shows how to color the tree: starting from the root, the layers of the tree are colored alternatingly with colors 1 (dark) and 2 (light). If there is no isolated vertex in the graph , then a weak 2-coloring determines a domatic partition: the set of the nodes with is a dominating set, and the set of the nodes with is another dominating set. ==Applications==

Historically, weak coloring served as the first non-trivial example of a graph problem that can be solved with a local algorithm (a distributed algorithm that runs in a constant number of synchronous communication rounds). More precisely, if the degree of each node is odd and bounded by a constant, then there is a constant-time distributed algorithm for weak 2-coloring. This is different from (non-weak) vertex coloring: there is no constant-time distributed algorithm for vertex coloring; the best possible algorithms (for finding a minimal but not necessarily minimum coloring) require communication rounds. Here is the iterated logarithm of . ==References==