More on the $k$-color connection number of a graph

An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors needed to color the edges of $G$ so that $G$ is $k$-color connected. First, we prove that let $H$ be a subdivision of a connected graph $G$, then $cc_{k}(H)\leq cc_{k}(G)$. Second, we give sufficient conditions to guarantee that $cc_{k}(G)=k$ in terms of minimum degree and the number of edges for 2-connected graphs. As a byproduct, we show that almost all graphs have the $k$-color connection number $k$. At last, we investigate the relationship between the $k$-color connection number and the rainbow connection number for a connected graph. In addition, we give exact values of $k$-color connection numbers for some graph classes: subdivisions of the wheel and the complete graph, and the generalised $\theta$-graph.