Every
forest is planar, and every planar graph can be partitioned into at most three forests. Therefore, the thickness of any graph 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 d have thickness at most \lceil d/2\rceil. This cannot be improved: for a d-regular graph with
girth at least 2d, the high girth forces any planar subgraph to be sparse, causing its thickness to be exactly \lceil d/2\rceil. of a 6-vertex
complete graph and 5-vertex cycle graph (center). Because the adjacencies in this graph are the union of those in two planar maps (left and right) it has thickness two. Graphs of thickness t with n vertices have at most t(3n-6) edges. Because this gives them average degree less than 6t, their
degeneracy is at most 6t-1 and their
chromatic number is at most 6t. Here, the degeneracy can be defined as the maximum, over subgraphs of the given graph, of the minimum degree within the subgraph. In the other direction, if a graph has degeneracy D then its arboricity and thickness are at most D. One can find an ordering of the vertices of the graph in which each vertex has at most D neighbors that come later than it in the ordering, and assigning these edges to D distinct subgraphs produces a partition of the graph into D trees, which are planar graphs. Even in the case t=2, the precise value of the chromatic number is unknown; this is
Gerhard Ringel's
Earth–Moon problem. An example of Thom Sulanke shows that, for t=2, at least 9 colors are needed. ==Related problems==