Acyclic colorings of planar graphs

It is shown that a planar graph can be partitioned into three linear forests. The sharpness of the result is also considered. In 1969, Chartrand and Kronk [2] showed that the vertex arboricity of a planar graph is at most 3. In other words, the vertex set of a planar graph can be partitioned into three sets each inducing a forest. In this paper we present an improvement on this result: that the vertex set of a planar graph can be partitioned into three sets such that each set induces a linear forest. A linear forest is one in which every component is a path. We will, for brevity, call such a partition a 3LF-coloring. This result establishes a conjecture of Broere and Mynhardt [1]. It also improves upon a result of Cowen, Cowen and Woodall [3] that one is guaranteed a partition into three sets each inducing a graph of maximum degree at most two. Our result is proved by a simple extension of the techniques in [3]. Theorem 1 The vertex set of any planar graph can be partitioned into three sets such that each set induces a linear forest. Proof. We prove by induction on the number of vertices that: every planar graph can be partitioned as above, with the added requirement that any two specified adjacent vertices are properly colored. This is true for graphs without edges. So let G be a planar graph with specified adjacent vertices u and v. If one of u and v has degree two or less, then a simple inductive argument will establish the result. So we may assume that u and v have degree at least three. Consider an embedding of G in the plane. Insert a new vertex x subdividing the edge uv. Further, while preserving planarity, add edges to ensure that there