In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in p...

We derive a general result concerning cyclic orderings of the elements of a matroid. As corollaries we obtain two further results. The first corollary proves a conjecture of Gonçalves [7], stating that the circular arboricity of a matroid is equal to its fractional arboricity. This generalises a well-known result from Nash-Williams on covering graphs by spanning trees, and a result from Edmonds...

The thickness of a graph G = (V,E) with n vertices is the minimum number of planar subgraphs of G whose union is G. A polyline drawing of G in R is a drawing Γ of G, where each vertex is mapped to a point and each edge is mapped to a polygonal chain. Bend and layer complexities are two important aesthetics of such a drawing. The bend complexity of Γ is the maximum number of bends per edge in Γ,...

We present a data stream algorithm for estimating the size of the maximum matching of a low arboricity graph. Recall that a graph has arboricity α if its edges can be partitioned into at most α forests and that a planar graph has arboricity α = 3. Estimating the size of the maximum matching in such graphs has been a focus of recent data stream research [1–3,5, 7]. See also [6] for a survey of t...

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(∆(G))/2e if G has no 5-cycles with chords. 2010 Mathematics Subject Classification: 05C15

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We prove that for an undirected graph with arboricity at most k+ǫ, its edges can be decomposed into k forests and a subgraph with maximum degree ⌈ 1−ǫ ⌉. The problem is solved by a linear programming based approach: we first prove that there exists a fractional solution to the problem, and then use a result on the degree bounded matroid problem by Király, Lau and Singh [5] to get an integral so...

A map from E (G) to {1, 2, 3, ..., t} is called a t-linear coloring if (V (G), φ(α)) is a linear forest for 1 ≤ α ≤ t. The linear arboricity la (G) of a graph G defined by Harary [9] is the minimum number t for which G has a t-linear coloring. Let G be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove that if G contains no 4-cycles and intersecting triangles...