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

如果边缘彩色图中的所有路径具有相同的颜色，则该路径称为单色路径。如果每个\（P_i \）是单色的并且对于任何两个\（i \ ne j \），\（P_i \）和\（P_j \），我们将k条路径称为（P_1，\ ldots，P_k \）彩虹单色路径。有不同的颜色。边缘着色图的ģ被说成是一个彩虹单色ķ _edge时连接的着色（或\（RMC_k \） -coloring的简称）如果每两个不同的顶点ģ通过至少连接ķ彩虹单色路径。我们使用\（rmc_k（G）\）表示确保G具有\（RMC_k \）着色的最大颜色数，该数字称为彩虹单色k边缘连接数。我们证明了\（RMC_k \） -图的着色的存在，然后给出了\（rmc_k（G）\）的一些界限，并给出了\（rmc_k（G）\）达到下界的一些图。我们还获得\（rmc_k（G（n，p））\ ge f（n）\）的阈值函数，其中\（\ left \ lfloor \ frac {n} {2} \ right \ rfloor> k \ ge 1 \）。