Monochromatic $k$-connection of graphs

An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of $G$ such that, any two vertices are connected by $k$ internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter $mc_k(G)$. We obtain bounds for $mc_k(G)$, for general graphs $G$. We also compute $mc_k(G)$ exactly when $k$ is small, and $G$ is a graph on $n$ vertices, with a spanning $k$-connected subgraph having the minimum possible number of edges, namely $\lceil\frac{kn}{2}\rceil$. We prove a similar result when $G$ is a bipartite graph.