An edge colored graph G is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this work we study the rainbow connectivity of the binomial graph G = G(n, p) at the connectivity threshold p = ...

This paper deals the graphs for which the removal of any edge changes the majority domination number of the graph. γM -critical edges, γM -redundant edges, γM -durable graphs and γM -critical graphs are studied. Further, majority domination critical edges and majority domination critical graphs are characterized.

In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is ...

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that • max rainbow matching can be approximated by a ...

A graph is k-domination-critical if γ(G)=k, and for any edge e not in G, γ(G+e) = k-1. In this paper we show that the diameter of a domination k-critical graph with k≥2 is at most 2k-2. We also show that for every k≥2, there is a k-domination-critical graph having diameter 3 2 k 1 −     . We also show that the diameter of a 4-domination-critical graph is at most 5.

A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. In this paper, we s...

The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a g...

Given a simple graph G = (V, E), an edge (u, u) E E is said to dominate itself and any edge (u,x) or (u,x), where x E V. A subset D C E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge e E E is associated with a r...

Abstract By suitably adjusting the tropical algebra technique we compute rainbow independent domination numbers of several infinite families graphs including Cartesian products $$C_n \Box P_m$$ C n ? P m </mm...