Let k be a positive integer and G be a k-connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k-connection number rck(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rck(G) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths...

A path in an edge-colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge-colored graph is (strongly) rainbow connected if there exists a rainbow (geodesic) path between every pair of vertices. The (strong) rainbow connection number of G, denoted by (scr(G), respectively) rc(G), is the smallest number of colors that are needed in order to make G (stro...

Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of Kn using precisely n − 1 colors, the edge set can be partitioned into n2 spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Kaneko, Kano and Suzuki improved this to three edg...

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...

A study was undertaken to compare the virulence and serum killing resistance properties of Atlantic salmon and rainbow trout Yersinia ruckeri isolates. Five isolates, covering heat-stable O-antigen O1, O2 and O5 serotypes, were tested for virulence towards fry and juveniles of both species by experimental bath challenge. The sensitivity of 15 diverse isolates to non-immune salmon and rainbow tr...

0  1  2  3   as does its derivative  0 (): 0  0 1   0 2   0 3     0 0 = 0 01   0 02   0 03   0 04    = 0 See Tables 1 & 2 for the cases  = 0 1 2 and Tables 3 & 4 for the cases  = 12 32 52. These appear in many physical applications that we cannot hope to survey in entirety. We will state only a few properties and several importan...

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 the path are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow kconnectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that any two distinct vertices of G are connected by k in...

A cycle in an edge-colored graph is said to be rainbow if no two of its edges have the same color. For a complete, infinite, edge-colored graph G, define S(G) = {n ≥ 2 | no n-cycle of G is rainbow}. Then S(G) is a monoid with respect to the operation n ◦m = n + m − 2, and thus there is a least positive integer π(G), the period of S(G), such that S(G) contains the arithmetic progression {N + kπ(...

A path in an edge colored graph is said to be a rainbow path if every edge in this colored with the same color. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertexconnection number of G denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainb...