A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f(δ) such that a properly edgecolored graph G with minimum degree δ and order at least f(δ) must have a rainbow matching of size δ. We answer this question in the affirmative; an extremal approach yields that f(δ) = 98δ/23 < 4.27δ suffices. Furthermore, we giv...

Let G = (V, E) be an undirected graph with a set V of nodes and a set E of edges, |V | = n. A node v is said to distance-k dominate a node w if w is reachable from v by a path consisting of at most k edges. A set D ⊆ V is said a distance-k dominating set if every node can be distance-k dominated by some v ∈ D. The size of a minimum distance-k dominating set, denoted by γk(G), is called the dist...

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

Given a function f a subset of its domain is a rainbow subset for f if f is one to one on it. We start with an old Erdős Problem: Assume f is a coloring of the pairs of ω1 with three colors such that every subset A of ω1 of size ω1 contains a pair of each color. Does there exist a rainbow triangle ? We investigate rainbow problems and results of this style for colorings of pairs establishing ne...

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pe...

Let G be a 2-connected graph. A subset D of V (G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove that γ2(G) ≤ n ln δ(G) δ(G) (1 + oδ(1)) where oδ(1) denotes a function that tends to 0 as δ →...

Connected Dominating Set is widely used as virtual backbone in wireless networks to improve network performance and optimize routing protocols. Based on special characteristics of ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacent if the distance bet...

Connected Dominating Set is widely used as virtual backbone in wireless Ad-hoc and sensor networks to improve the performance of transmission and routing protocols. Based on special characteristics of Ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacen...