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. For any two vertices u and v of G, a rainbow u− v geodesic in G is a rainbow u− v path of length d(u, v), where d(u, v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u − v geodesic for any two vertices...

BACKGROUND Despite vaccination with a commercial vaccine with a documented protective effect against Vibrio anguillarum O1 disease outbreaks caused by this bacterium have been registered among rainbow trout at Danish fish farms. The present study examined specific serum antibody levels as a valid marker for assessing vaccination status in a fish population. For this purpose a highly sensitive e...

This article contains proofs of Theorems 1.1, 1.2, 1.3 and 1.4 in [10].

Let p be a prime number and Zp be the cyclic group of order p. A coloring of Zp is called rainbow–free with respect to a certain equation, if it contains no rainbow solution of the equation, that is, a solution whose elements have pairwise distinct colors. In this paper we describe the structure of rainbow–free 3–colorings of Zp with respect to all linear equations on three variables. Consequen...

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex v is the number of different colors on edges incident to v. Wang and Li conjectured that for k > 4, every edge-colored graph with minimum color degree at least k contains a rainbow matching of size at least ⌈k/2⌉. We prove the slightly weaker statement that a rainbow matching...

We consider the MINIMUM RAINBOW SUBGRAPH problem (MRS): Given a graph G whose edges are coloured with p colours. Find a subgraph F ⊆ G of minimum order and with p edges such that each colour occurs exactly once. This problem is NP-hard and APX-hard. For a given graph G and an edge colouring c with p colours we define the rainbow subgraph number rs(G, c) to be the order of a minimum rainbow subg...

Time memory tradeoff algorithms are tools for inverting oneway functions, and they are often used to recover passwords from unsalted password hashes. There are many publicly known tradeoff algorithms, and the rainbow tradeoff is widely believed to be the best algorithm. This work provides an accurate complexity analysis of the fuzzy rainbow tradeoff algorithm, which has not yet received much at...

Rainbow vertex-connection number is the minimum colors assignment to vertices of graph, such that each vertex connected by a path whose edges have distinct and denoted . The rainbow connection can be applied graphs resulting from operations. One methods create new graph perform operations between two graphs. Thus, this research uses comb product operation determine rainbow-vertex cycle complete...