2.9 Coloring & Connectivity

Graph Coloring II

What is the chromatic number of an acyclic graph?

Exercise 1

By definition, an acyclic graph contains no cycles (and in particular, no cycles of odd length). Therefore it is bipartite. This means 2 colors are sufficient. Moreover, if the graph has at least one edge, 2 colors are also necessary. Therefore, the chromatic number of any acyclic graph is 2.

Note that the answer assumes the non-trivial case, where the graph contains at least one edge. In the trivial case where the graph contains no edges, it only has disjoint vertices and its chromatic number is 1.