[show abstract][hide abstract] ABSTRACT: . The acyclic list chromatic number of every 1-planar graph is proved to be at most 7 and is conjectured to be at most 5. Keywords. Acyclic coloring, List coloring, Acyclic list coloring. 1
[show abstract][hide abstract] ABSTRACT: 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 pla- nar 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 im- provement 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).
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