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For a nontrivial connected graph $G$ of order $n$, the detour distance $D(u,v)$ between two vertices $u$ and $v$ in $G$ is the length of a longest $u-v$ path in $G$. Detour distance is a metric on the vertex set of $G$. For each integer $k$ with $1\le k\le n-1$, a coloring $c:V\left(G\right)\to \mathbb{N}$ is a $k$-metric coloring of $G$ if $\left|c\right(u)-c(v\left)\right|+D(u,v)\ge k+1$ for every two distinct vertices $u$ and $v$ of $G$. The value ${\chi }_m^k\left(c\right)$ of a $k$-metric coloring $c$ is the maximum color assigned by $c$ to a vertex of $G$ and the $k$-metric chromatic number ${\chi }_m^k\left(G\right)$ of $G$ is the minimum value of a $k$-metric coloring of $G$. For every...