It is known that every planar graph $G$ can be decomposed into at most 5 spanning star forests, which means that there exists at most five edge-disjoint spanning subgraphs of $G$ each of which is a forest with connected components being stars.


My question is as follows.
Does there exist a constant $k$ such that every planar graph can be decomposed into at most $k$ spanning star forests such that every vertex of the graph is a center of at most one non-trivial (i.e., with at least 2 vertices) star in the decomposition?



Decomposing planar graph into star forests with non-intersecting centers
There is such a constant, namely $k=5$. 


The proof that there is a decomposition into five star forests in [1] is based on an acyclic $5$-colouring of the vertex set. A vertex $v$ cannot be the centre of a non-trivial star in the forest $F\_i$ unless it receives colour $i$ in the colouring. In particular, no vertex will be the centre of non-trivial stars in two different forests. 


[1] *Hakimi, S.L.; Mitchem, J.; Schmeichel, E.*, [**Star arboricity of graphs**](http://dx.doi.org/10.1016/0012-365X(94)00313-8), Discrete Math. 149, 1996.


