Reading Rainbow

Rainbow Matchings and Rainbow Connectedness

Aharoni and Berger conjectured that every collection of n matchings of size n+1 in a bipartite graph contains a rainbow matching of size n. This conjecture is related to several old conjectures of Ryser, Brualdi, and Stein about transversals in Latin squares. There have been many recent partial results about the Aharoni-Berger Conjecture. The conjecture is known to hold when the matchings are m...

Rainbow.

Rainbow edge-coloring and rainbow domination

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′(...

Rainbow Decompositions

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson’s Theorem. For every integer k there exists an n0 = n0(k) so that for all n > n0, if n mod k(k − 1) ∈ {1, k}, then every properly edge-colored Kn contains (n 2 ) / (k 2 ) pairwise edge-disjoint rainbow copies of Kk. Our proof uses, as a main ingredient, a double application ...

Polarized rainbow.

The Airy theory of the rainbow is extended to polarized light. For both polarization directions a simple analytic expression is obtained for the intensity distribution as a function of the scattering angle in terms of the Airy function and its derivative. This approach is valid at least down to droplet diameters of 0.3 mm in visible light. The degree of polarization of the rainbow is less than ...