Albertson and Berman [1] conjectured that every planar graph has an induced forest on half of its vertices; the current best result, due to Borodin [3], is an induced forest on two fifths of the vertices. We show that the Albertson-Berman conjecture holds, and is tight, for planar graphs of treewidth 3 (and, in fact, for any graph of treewidth at most 3). We also improve on Borodin’s bound for ...

We give new bounds on the star arboricity and the caterpillar arboricity of planar graphs with given girth. One of them answers an open problem of Gyárfás and West: there exist planar graphs with track number 4. We also provide new NP-complete problems.

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...

A star is an arborescence in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. The directed star arboricity of a digraph D, denoted by dst(D), is the minimum number of galaxies needed to cover A(D). In this paper, we show that dst(D) ≤ Δ(D) + 1 and that if D is acyclic then dst(D) ≤ Δ(D). These results are proved by considering the existence of spann...

Let M be a closed 2-manifold. The chromatic number of M is defined to be the maximum chromatic number of all graphs which can be imbedded in M. The famous Four Colour Conjecture states that the chromatic number of the sphere is four. One of the oddities of mathematics is that the chromatic number of the familiar sphere is still unknown, although the chromatic number of every other closed 2-mani...

A directed star forest is a forest all of whose components are stars with arcs emanating from the center to the leaves. The acircuitic directed star arboricity of an oriented graph G (that is a digraph with no opposite arcs) is the minimum number of edge-disjoint directed star forests whose union covers all edges of G and such that the union of any two such forests is acircuitic. We show that e...

We prove that for any positive integer k and for ε = 1/((k + 2)(3k2 + 1)), the edges of any graph whose fractional arboricity is at most k + ε can be decomposed into k forests and a matching.

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, we completely determine lak(G) when G is a balanced complete bipartite graph Kn,n or a complete graph Kn, and k = 3. © 2007 Elsevier B.V. All rights reserved.

It is proved that the linear arboricity of every 1-planar graph with maximum degree ∆ > 33 is ⌈∆/2⌉.