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Then there is no way to color the vertices without ever finding that both endpoints of some edge have the same color. On the other hand, if we have four colors available then we can do the job. Fig. 4 There are many interesting computational and theoretical problems in the area of coloring of graphs. Just for its general interest, we are going to mention the four-color theorem, and then we will turn to a study of some of the computational aspects of graph coloring. First, just for general cultural reasons, let’s slow down for a while and discuss the relationship between graph colorings in general and the four-color problem, even though it isn’t directly relevant to what we’re doing.

C) A tree is a graph G with the property that between every pair of distinct vertices there is a unique path. If G is a graph and S ⊆ V (G), then S is an independent set of vertices of G if no two of the vertices in S are adjacent in G. An independent set S is maximal if it is not a proper subset of another independent set of vertices of G. Dually, if a vertex subset S induces a complete graph, then we speak of a complete subgraph of G. A maximal complete subgraph of G is called a clique. A graph might be labeled or unlabeled.

For a graph G we can define a number γ(G) = |V (G)| + |E(G)|, which is rather an odd kind of thing to define, but it has a nice property with respect to this algorithm, namely that whatever G we begin with, we will find that γ(G − {e}) = γ(G) − 1; γ(G/{e}) ≤ γ(G) − 2. 8) Indeed, if we delete the edge e then γ must drop by 1, and if we collapse the graph on the edge e then we will have lost one vertex and at least one edge, so γ will drop by at least 2. 9) then we claim that h(γ) ≤ h(γ − 1) + h(γ − 2) (γ ≥ 2).

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