Improper Colourings of Unit Disk Graphs

The arboricity a ( G ) of a graph G is the minimum number of forests needed to cover the edges of G. For a tree T, we similarly define the T-free arboricity T f a ( G ) of G as the minimum number of T-free forests needed to cover the edges of G. We give bounds on the maximum T-free arboricity of a planar graph with girth g for various T and g. One of them solves an open problem of Gyárfás and West: there exist planar graphs with track number 4. We also provide new NP-complete problems in sparse planar graphs. A very elementary one is the problem for any g ≥ 4 to determine if a planar bipartite graph with maximum degree three and girth at least g has star arboricity two.