We prove that the thickness and the arboricity of a graph with e edges are at most Lfl3 + 3/2J and r~l, respectively, and that the laller bound is best possible. The thickness of a graph G, e(G), is the mInImum number of planar graphs into which the edges of G can be partitioned, and the arboricity, J(G), is the minimum number of acyclic graphs into which the edges of G can be partitioned. Nash...

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 is the minimum number of edge-disjoint directed star forests whose union covers all edges of G and such that the union of two such forests is acircuitic. We show that graphs with maximum average degree less than 7 ...

Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G′ be the graph obtained from G...

The Goldberg–Seymour Conjecture for f $f$ -colorings states that the -chromatic index of a loopless multigraph is essentially determined by either weighted maximum degree or density parameter. We introduce an oriented version -colorings, where now each color class edge-coloring required to be orientable in such way every vertex v $v$ has indegree and outdegree at most some specified values g ( ...

We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs so long as a standard separator theorem exists. For example, we can apply it to show the known bo...

α-acyclicity is an important notion in database theory. The α-arboricity of a hypergraphH is the minimum number of α-acyclic hypergraphs that partition the edge set of H. The α-arboricity of the complete 3-uniform hypergraph is determined completely.

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that for a planar graph G with maximum degree ∆(G) ≥ 7, la(G) = d 2 e if G has no 5-cycles with chords.

In a directed graph, a star is an arborescence with at least one arc, in which the root dominates all the other vertices. A galaxy is a vertex-disjoint union of stars. In this paper, we consider the Spanning Galaxy Problem of deciding whether a digraph D has a spanning galaxy or not. We show that although this problem is NP-complete (even when restricted to acyclic digraphs), it becomes polynom...

1. Introduction Constrained orientations, that is orientations such that all the vertices have a prescribed indegree, relates to one another many combinatorial and topological properties such as arboricity, connectivity and planarity. These orientations are the basic tool to solve planar augmentation problems 2]. We are concerned with two classes of planar graphs: maximal planar graphs (i.e. po...