Arboricity and Chromatic number are linked in various ways. The topic also links to difficult NP complete problems. We muse about how often it is the case that for manifolds the question is easy. An example is the Hamiltonian path problem which is linked to Peg Solitaire
We aim to write down a short proof of the statement that a planar graph has arboricity 3 or less.
We explain why the arboricity of 3 spheres can take values between 4 and 7 and mention that for 3 manifolds the upper bound is 9 (but believed to be 7).
We look at the problem to find the possible arboricities which a 3-sphere can have.
We work on the result that any 2-sphere can be covered by 3 trees.
In other words: three trees suffice.
The three tree theorem tells that any discrete 2-sphere has arboricity exctly 3.
We aim to show that the Lusternik-Schnirelmann category of a graph is bound by the augmented dimension. We can try to prove this by using tree coverings and so look at arboricity.
