We shall first look at a few so-called coloring problems for graphs.

A proper coloring of a graph G is a function from the vertices to a set C of ‘colors’ (e.g. C = {1, 2, 3, 4}) such that the ends of every edge have distinct colors. (So a graph with a loop will admit no proper colorings.) If |C| = k, we say that G is k-colored.

The chromatic number χ(G) of a graph G is the minimal number of colors for which a proper coloring exists.

If χ(G) = 2 (or χ(G) = 1, which is the case when and only when G has no edges), then G is called bipartite. A graph with no odd polygons (equivalently, no closed paths of odd length) is bipartite as the reader should verify.

The famous ‘Four Color Theorem’ (K. Appel and W. Haken, 1977) states that if G is planar, then χ(G) ≤ 4.

Clearly χ(Kn) = n. If k is odd then χ(Pk) = 3. In the following theorem, we show that, with the exception of these examples, the chromatic number is at most equal to the maximum degree (R. L. Brooks, 1941).

Theorem 3.1.Let d ≥ 3 and let G be a graph in which all vertices have degree ≤ d and such that Kd+1 is not a subgraph of G. Then χ(G) ≤ d.

Proof 1: As is the case in many theorems in combinatorial analysis, one can prove the theorem by assuming that it is not true, then considering a minimal counterexample (in this case a graph with the minimal number of vertices) and arriving at a contradiction.