Lecturer : Alistair Sinclair Scribes : Francis Bach ,

We can do so by treating the vertices one by one. Consider the intermediate situation where V1 is the set of vertices that have already got their final color (i.e., the color in X2) and V2 = V \V1. Now, pick some vertex v ∈ V2. This vertex still has the color c1 it has in state X1 and needs to be recolored to the color c2 it is supposed to have in state X2. This is only possible if the neighbors of v have a color different of c2. Therefore we have to recolor first all neighbors of v that have color c2 (note that neighbors of v in V1 cannot have color c2 because X2 is a legal coloring). To recolor a neighbor w ∈ V2 of v, we have at least q − 1−∆ colors available (q colors in total, minus c2, minus all colors of neighbors of w). Since q ≥ ∆+ 2, we have at least one color available to color w. So we can recolor all v’s offending neighbors and then color v itself with c2. This can be repeated until finally V1 = V and we reach X2.