Graph coloring , perfect graphs 1 Introduction to graph coloring

Let us improve this bound. Assume that G is a connected graph and T is its spanning tree rooted at r. Let us consider an ordering of V (G) in which each vertex v appears after its children in T . Now, for v 6= r we have |N(vi) ∩ {v1, . . . , vi−1}| ≤ deg v − 1, so c(vi) ≤ deg vi for vi 6= r. Unfortunately, the greedy may still need to use ∆(G) + 1 colors if deg r = ∆(G) and each child of r happens to be colored using a different color. Nevertheless, we have a lot of freedom in choosing T and it turns out that for each graph we can either find an appropriate tree T or prove that no ∆(G)-coloring exists.