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...

Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

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...

An Italian dominating function on a (di)graph G with vertex set V ( ) is f : → { 0 , 1 2 } such that every v ∈ = has an (in)neighbour assigned or two (in)neighbours 1. We complete the investigation of domination numbers Cartesian products directed cycles.

A new class of rainbows is created when a droplet is illuminated from the inside by a point light source. The position of the rainbow depends on both the index of refraction of the droplet and the position of the light source, and the rainbow vanishes when the point source is too close to the center of the droplet. Here we experimentally measure the position of the transmission and one-internal...

An edge-colored graph is rainbow if all its edges are colored with distinct colors. For a fixed graph H , the rainbow Turán number ex(n,H) is defined as themaximumnumber of edges in a properly edge-colored graph on n vertices with no rainbow copy of H . We study the rainbow Turán number of even cycles, and prove that for every fixed ε > 0, there is a constant C(ε) such that every properly edge-...