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 ...

Let C = (C1, . . . , Cm) be a system of sets. The range of an injective partial choice function from C is called a rainbow set, and is also said to be multicolored by C. If φ is such a partial choice function and i ∈ dom(φ) we say that Ci colors φ(i) in the rainbow set. If the elements of Ci are sets then a rainbow set is said to be a (partial) rainbow matching if its range is a matching, namel...

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...

We present the Rainbow Code, a state-of-the-art genetic algorithm (GA) representation that encodes layered trees or bipartite trees (spanning trees of a pre-specified complete layered graph or complete bipartite graph) as integer strings of a certain form. The Rainbow Code possesses perfect feasibility, full coverage and zero bias, because the correspondence it defines between trees and strings...

Hybridization between rainbow trout (Oncorhynchus mykiss (Walbaum, 1792)) and westslope cutthroat trout (Oncorhynchus clarkii lewisi (Girard, 1856)) occurs commonly when rainbow trout are introduced into the range of westslope cutthroat trout. Typically, hybridization ismost common inwarmer, lower elevation habitats, butmuch less common in colder, higher elevation habitats. We assessed the tole...

The first likely photographic observation of the tertiary rainbow caused by sunlight in the open air is reported and analyzed. Whereas primary and secondary rainbows are rather common and easily seen phenomena in atmospheric optics, the tertiary rainbow appears in the sunward side of the sky and is thus largely masked by forward scattered light. Up to now, only a few visual reports and no relia...

The rainbow game domination subdivision number of a graph G is defined by the following game. Two players D and A, D playing first, alternately mark or subdivide an edge of G which is not yet marked nor subdivided. The game ends when all the edges of G are marked or subdivided and results in a new graph G′. The purpose of D is to minimize the 2-rainbow dominating number γr2(G ′) of G′ while A t...

Let G be an edge-colored graph with n vertices. A rainbow subgraph is a subgraph whose edges have distinct colors. The rainbow edge-chromatic number of G, written χ̂′(G), is the minimum number of rainbow matchings needed to cover E(G). An edgecolored graph is t-tolerant if it contains no monochromatic star with t+1 edges. If G is t-tolerant, then χ̂′(G) < t(t+ 1)n lnn, and examples exist with χ̂′(...

A 3–coloring of an abelian group G is rainbow–free if there is no 3–term arithmetic progression with its members having pairwise distinct colors. We describe the structure of rainbow–free colorings of abelian groups. This structural description proves a conjecture of Jungić et al. on the size of the smallest chromatic class of a rainbow–free coloring of cyclic groups.

A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of n-rainbow graphs on 2n vertices and the switching classes of graphs on n vertices.