Thickness (graph theory)

In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k. In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G. Thus, a planar graph has thickness 1. Graphs of thickness 2 are called biplanar graphs. The concept of thickness originates in the Earth–Moon problem on the chromatic number of biplanar graphs, posed in 1959 by Gerhard Ringel, and on a related 1962 conjecture of Frank Harary: For any graph on 9 points, either itself or its complementary graph is non-planar. The problem is equivalent to determining whether the complete graph K9 is biplanar (it is not, and the conjecture is true). A comprehensive survey on the state of the arts of the topic as of 1998 was written by Petra Mutzel, Thomas Odenthal and Mark Scharbrodt. The thickness of the complete graph on n vertices, Kn, is except when n = 9, 10 for which the thickness is three. With some exceptions, the thickness of a complete bipartite graph Ka,b is generally: Every forest is planar, and every planar graph can be partitioned into at most three forests. Therefore, the thickness of any graph G is at most equal to the arboricity of the same graph (the minimum number of forests into which it can be partitioned) and at least equal to the arboricity divided by three. The graphs of maximum degree have thickness at most . This cannot be improved: for a -regular graph with girth at least , the high girth forces any planar subgraph to be sparse, causing its thickness to be exactly . Graphs of thickness with vertices have at most edges. Because this gives them average degree less than , their degeneracy is at most and their chromatic number is at most . Here, the degeneracy can be defined as the maximum, over subgraphs of the given graph, of the minimum degree within the subgraph.