@@ -45,3 +45,19 @@ Assign a set from the cover free sets to each of the nodes. Then if every node l

To further reduce the colors needed to $`\Delta+1`$, the colors are grouped into groups with size $`2\Delta+2`$. Then in every round the node with the highest color in the group chooses a color from the lower half of the group. Different groups do this in parallel, since they cannot choose the same color.

If one iteration is complete, the remaining colors are regrouped again in the same way, but now there are only half as many groups. This is done until only one group is left.

# 5. Lecture

## Maximal Independence Sets

A subset of nodes of a graph, where no to nodes are neighbors.

## MIS coloring

Copy the graph $`\Delta+1`$ times and connect all vertexes that where copied from an original vertex in full mesh, e.g. $`w, w', w'', w'''`$. Then construct a MIS. Then color every original node with the number of the copy, the vertex chosen by the MIS exists in.

## Luby's Algorithm

1. Assign a random number to all vertexes.

2. If a vertex's number is strictly larger than the ones of its neighbors, remove all neighbors.

3. Repeat from 1. until no reduction has to be made anymore.