Rainbow Monochromatic k-Edge-Connection Colorings of Graphs

Abstract

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 $\left\lfloor \frac{n}{2}\right\rfloor > k\ge 1$.