A linear k-forest is a graph whose components are paths of length at most k. The linear k-arboricity of a graph G, denoted by lak(G), is the least number of linear k-forests needed to decompose G. In this paper, it is obtained that la4(Kn,n) = d5n/8e for n ≡ 0( mod 5).

Chung and Liu defined the d-chromatic Ramsey numbers as a generalization of Ramsey numbers by replacing a weaker condition. Let 1 < d < c and let t = (c d ) . Assume A1, A2, . . . , At are all d-subsets of a set containing c distinct colors. Let G1, G2, . . . , Gt be graphs. The d-chromatic Ramsey number denoted by rc d(G1, G2, . . . , Gt) is defined as the least number p such that, if the edge...

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ vertices graph $G$ is \emph{distance-edge-monitoring set} if for every edge $e$ $G$, there vertex $x$ and $y$ such that belongs to all shortest paths between $y$. denote by $dem(G)$ smallest size $G$. The represent distance probes modeled $G$; when fails, from increases, thus we are able detect failure. It t...

If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is ∆ if ∆ ≥ 8, the list edge ...

Let G be a planar graph with maximum degree ∆ ≥ 5. It is proved that la(G) = d∆(G)/2e if (1) any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4, 5} or (2) G has no intersecting 4-cycles and intersecting i-cycles for some i ∈ {3, 6}. 2010 Mathematics Subject Classification: 05C15

A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerpla...

The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of ...