A linear forest is the union of a set of vertex disjoint paths. Akiyama, Exoo and Harary and independently Hilton have conjectured that the edges of every graph of maximum degree ∆ can be covered by d∆+1 2 e linear forests. We show that almost every graph can be covered with this number of linear forests. 1 Linear Forests, Directed Cycles, and the Probabilistic Method A linear forest is the uni...

A linear k-forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity Of G, denoted by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In case that the lengths of paths are not restricted, we then have the linear arboricity ofG, denoted by la(G). In this paper, the exact values of th...

A k-bar visibility representation of a digraph G assigns each vertex at most k horizontal segments in the plane so that G has an arc uv if and only if some segment for u “sees” some segment for v above it by a vertical line of sight. The (bar) visibility number b(G) of a digraph G is the least k permitting such a representation. Among other results, we show that b(G) ≤ 4 when G is a planar digr...

A graph is a linear forest if each of its components path. Given G with maximum degree Δ(G), motivated by the famous arboricity conjecture and Lovász's classic result on partitioning edge set into paths, we call partition F:=F1|⋯|Fk an exact Fi induces forest, k≤⌈Δ(G)+12⌉, every vertex v∈V(G) at most ⌈dG(v)+12⌉ non-trivial paths belonging to F. In this paper, prove following two results. Every ...

A linear forest is a in which every connected component path. The arboricity of graph G the minimum number forests covering all edges. In 1980, Akiyama, Exoo, and Harary proposed conjecture, known as Linear Arboricity Conjecture (LAC), stating that Δ-regular has ⌈ Δ + 1 2 ⌉ . 1988, Alon proved LAC holds asymptotically. 1999, list version was raised by An Wu, called List Conjecture. this article...

In a linear forest, each component is a path. The linear arboricity ~(G) of a graph G is defined in Harary [8] as the minimum number of linear forests whose union is G. This invariant first arose in a study [i0] of information retrieval in file systems. A quite similar covering invariant which is well known to the linear arboricity is the arboricity of a graph, which is defined as the minimum n...

An equitable k-coloring of a graph G is proper such that the sizes any two color classes differ by at most one. (q,r)-tree-coloring an q-coloring subgraph induced each class forest maximum degree r. Let strong vertex r-arboricity G, denoted var≡(G), be minimum p has for every q≥p. The values va1≡(Kn,n) were investigated Tao and Lin Wu, Zhang, Li where exact found in some special cases. In this ...

We revisit the classic problem of estimating the degree distribution moments of an undirected graph. Consider an undirected graph G = (V,E) with n (non-isolated) vertices, and define (for s > 0) μs = 1 n · ∑ v∈V d s v. Our aim is to estimate μs within a multiplicative error of (1 + ε) (for a given approximation parameter ε > 0) in sublinear time. We consider the sparse graph model that allows a...

The arboricity of a hypergraph H is the minimum number of acyclic hypergraphs that partition H . The determination of the arboricity of hypergraphs is a problem motivated by database theory. The exact arboricity of the complete k-uniform hypergraph of order n is previously known only for k ∈ {1, 2, n − 2, n − 1, n}. The arboricity of the complete k-uniform hypergraph of order n is determined as...

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [1] stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum degree ∆ is either ⌈ ∆ 2 ⌉