An edge-colored graph G is rainbow connected if every two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph G, denoted by rcðGÞ, is the smallest number of colors that are needed in order to make G rainbow connected. It was proved that computing rcðGÞ is an NP-hard problem, as well as that even deciding whether a graph has rcðGÞ...

Let D be the minimum dominating set of intuitionistic fuzzy graph G. The minimum intuitionistic fuzzy cardinality of all edge dominating set of intuitionistic fuzzy graph G is known as edge domination number and it is denoted by γe(G). In this Paper, we initiate some definitions onedge dominating set concerning intuitionistic fuzzy sets. Further, we investigate some results onedge domination nu...

For positive integers k and d such that 4 ≤ k < d and k 6= 5, we determine the maximum number of rainbow colored copies of C4 in a k-edge-coloring of the d-dimensional hypercube Qd. Interestingly, the k-edge-colorings of Qd yielding the maximum number of rainbow copies of C4 also have the property that every copy of C4 which is not rainbow is monochromatic.

Let G be an edge-colored copy of Kn, where each color appears on at most n/2 edges (the edgecoloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth [4] conjectured that every properly edge-colored Kn (n ≥ 6 and even) using exactly n−1 colors has n/2 edge-disjoint rainbow spanning trees, and they proved...

In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge coloring of the complete graph Kn, there is a Hamilton cycle with at most √ 8n different colors. We also prove that in every proper edge coloring of the complete graph Kn, there is a rainbow cycle with at least n/2−1 colors (A rainbow cycle is a cycle whose all edges have different colors). We show ...

Let G = (V, E) be a graph without isolated vertices. A secure edge dominating set of G is an edge dominating set F⊆E with the property that for each e ∈ E – F, there exists f∈F adjacent to e such that (F – {f}) ∪ {e} is an edge dominating set. The secure edge domination number γ's(G) of G is the minimum cardinality of a secure edge dominating set of G. In this paper, we initiate a study of the ...

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a 2-dominating set of G if S dominates every vertex of V (G) \ S at least twice. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-domination subdivision number sdγ2(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in ...

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of G, denoted by rc(G), is the minimum number of colors that are needed in order to make G rainbow connected. Char...

A set D of vertices of a graph G is a dominating set if every vertex in V \D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the domination number. Arumugam h...