A linear forest is a forest in which each connected component is a path. The linear arboricity la(G) of a graph G is the minimum number of linear forests whose union is the set of all edges of G. The linear arboricity conjecture asserts that for every simple graph G with maximum degree A = A(G), Although this conjecture received a considerable amount of attention, it has been proved only for A ...

We present two fully dynamic algorithms for maximum cardinality matching in bipartite graphs. Our main result is a deterministic algorithm that maintains a (3/2+ ) approximation in worst-case update time O(m −2.5), which is polynomially faster than all previous deterministic algorithms for any constant approximation, and faster than all previous algorithms (randomized included) that achieve a b...

Abstract We describe a simple deterministic $$O( \varepsilon ^{-1} \log \Delta )$$ O ( ε - 1 log Δ ) round distributed algorithm for $$(2\alpha +...

let g be a graph with p vertices and q edges and a = {0, 1, 2, . . . , [q/2]}. a vertex labeling f : v (g) → a induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. for a ∈ a, let vf (a) be the number of vertices v with f(v) = a. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the in...

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let g(H) = |E(H)| |V(H)|−ω(H) . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all nontrivial subgraphs H of G, is attained when H = G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensi...

A graph G is called H-saturated if G contains no copy of H, but for any edge e in the complement of G, the graph G + e contains some copy of H. The minimum size of an n-vertex H-saturated graph is denoted by sat(n,H) and is called the saturation number of H. In [?], Kászonyi and Tuza determined the values of sat(n,H) when H is a path or a disjoint union of edges. In this paper, we determine the...

We present a simple single-pass data stream algorithm using O( −2 logn) space that returns a (α+ 2)(1 + ) approximation to the size of the maximum matching in a graph of arboricity α. 1998 ACM Subject Classification F.2 Analysis of Algorithms & Problem Complexity

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e an edge incident to v. Two incidences (v, e) and (w, f) are adjacent whenever v = w, or e = f , or vw = e or f . The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980–1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alte...