Article history: The ability of rainbow trou Received 21 August 2007 Received in revised form 10 March 2008 Accepted 11 March 2008

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edgecoloring of G such that every two distinct vertices of G are connected by k interna...

We prove several results regarding edge-colored complete graphs and rainbow cycles, cycles with no color appearing on more than one edge. We settle a question posed by Ball, Pultr, and Vojtěchovský [BPV05] by showing that if such a coloring does not contain a rainbow cycle of length n, where n is odd, then it also does not contain a rainbow cycle of length m for all m greater than 2n. In additi...

Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition ( X , Y ) and . We show that if   = G k   3   7 max , 4 k X Y  , then G has a rainbow coloring of size at least 3 4 k       .

A path in an edge-colored graph is rainbow if no two edges of it are colored the same. The graph is said to be rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is strong rainbow connected. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or stro...

A path in an edge-colored graph G is rainbow if no two edges of it are colored the same. The graph G is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph G is strongly rainbow-connected. The minimum number of colors needed to make G rainbow-connected is known as the rainbow connection number...

Multivariate Public Key Cryptosystems (MPKC) is one of candidates for post-quantum cryptography. Rainbow is an MPKC digital signature scheme, with relatively efficient encryption and decryption processes. However, the size of the secret key of Rainbow is substantially larger than that of an RSA cryptosystem for the same security level. By using sparse secret keys, the size of the secret key of ...

For a finite simple edge-colored connected graph G (the coloring may not be proper), a rainbow path in G is a path without two edges colored the same; G is rainbow connected if for any two vertices of G, there is a rainbow path connecting them. Rainbow connection number, rc(G), of G is the minimum number of colors needed to color its edges such that G is rainbow connected. Chakraborty et al. (2...

Laboratory and in-stream enclosure experiments were used to determine whether rainbow trout Oncorhynchus mykiss influence survival of longnose dace Rhinichthys cataractae. In the laboratory, adult rainbow trout preyed on longnose dace in 42% of trials and juvenile rainbow trout did not prey on longnose dace during the first 6 h after rainbow trout introduction. Survival of longnose dace did not...

Let H (k) n,p,κ denote a randomly colored random hypergraph, constructed on the vertex set [n] by taking each k-tuple independently with probability p, and then independently coloring it with a random color from the set [κ]. Let H be a k-uniform hypergraph of order n. An `-Hamilton cycle is a spanning subhypergraph C of H with n/(k−`) edges and such that for some cyclic ordering of the vertices...