Monochromatic $k$-edge-connection colorings of graphs

Abstract

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths colored with a same color. We use $mc_k(G)$ and $umc_k(G)$ to denote the maximum number of colors that ensures $G$ to be monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly monochromatic $k$-edge-connected. In this paper, we first conjecture that for any $k$-edge-connected graph $G$, $mc_k(G)=e(G)-e(H)+\lfloor\frac{k}{2}\rfloor$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$. We verify the conjecture for $k=2$. We also prove the conjecture for $G=K_{k+1}$ when $k\geq4$ is even, and for $G=K_{k,n}$ when $k\geq4$ is even, or when $k=3$ and $n\geq k$. When $G$ is a minimal $k$-edge-connected graph, we give an upper bound of $mc_k(G)$, i.e., $mc_k(G)\leq k-1$, and $mc_k(G)\leq \lfloor\frac{k}{2}\rfloor$ when $G=K_{k,n}$. For the uniformly monochromatic $k$-edge-connectivity, we prove that for all $k$, $umc_k(G)=e(G)-e(H)+1$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$.