Rainbow Monochromatic k-Edge-Connection Colorings of Graphs

A path in an edge-colored graph is called a monochromatic if all edges of the have same color. We call k paths \(P_1,\ldots ,P_k\) rainbow every \(P_i\) and for any two \(i\ne j\), \(P_j\) different colors. An edge-coloring G said to be k-edge-connection coloring (or \(RMC_k\)-coloring short) distinct vertices are connected by at least paths. use \(rmc_k(G)\) denote maximum number colors that ensures has \(RMC_k\)-coloring, this number. prove existence \(RMC_k\)-colorings graphs, then give some bounds present graphs whose reaches lower bound. also obtain threshold function \(rmc_k(G(n,p))\ge f(n)\), where \(\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1\).