Abstract In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph G of maximum degree $$\Delta $$ Δ can be decomposed into at most "Equation missing" linear forests. (A forest is if all its components are paths.) 1988, Alon p...

Estimating the size of the maximum matching is a canonical problem in graph algorithms, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. • (Insert-Only Streams) We present a one-pass algorithm that takes O(c log n) space and a...

In 1999, Brodal and Fagerberg (BF) gave an algorithm for maintaining a low outdegree orientation of a dynamic uniformly sparse graph. Specifically, for a dynamic graph on n-vertices, with arboricity bounded by α at all times, the BF algorithm supports edge updates in O(log n) amortized update time, while keeping the maximum outdegree in the graph bounded by O(α). Such an orientation provides a ...

Let G = (V,E) be a graph with maximum degree k ≥ 3 distinct from Kk+1. Given integers s ≥ 2 and p1, . . . , ps ≥ 0, G is said to be (p1, . . . , ps)-partitionable if there exists a partition of V into sets V1, . . . , Vs such that G[Vi] is pi-degenerate for i ∈ {1, . . . , s}. In this paper, we prove that we can find a (p1, . . . , ps)-partition of G in O(|V |+|E|)-time whenever 1 ≥ p1, . . . ,...

Let G be a non-trivial, loop-less multi-graph and for each non-trivial sub-graph H of G, let g(H) = |E(H)| |V (H)|−ω(G) . G is said to be uniformly dense if and only if γ(G), the maximum among g(H) taken over all non-trivial subgraphs H of G is attained when H = G. This quantity γ(G) is called the fractional arboricity of G and was introduced by Catlin, Grossman, Hobbs and Lai [4]. γ(G)− g(G) m...

A digraph is m-labelled if every arcs is labelled by an integer in {1, . . . , m}. Motivated by wavelength assignment for multicasts in optical star networks, we study n-fiber colourings of labelled digraph which are colourings of the arcs of D such that at each vertex v, for each colour in λ, in(v, λ) + out(v, λ) ≤ n with in(v, λ) the number of arcs coloured λ entering v and out(v, λ) the numb...

We consider graph compression in terms of graph families. In particular, we show that graphs of bounded genus can be compressed to O(n) bits, where n is the number of vertices. We identify a property based on separators that makes O(n)-bit compression possible for some graphs of bounded arboricity.