Hardness result for the total rainbow $k$-connection of graphs

A path in a total-colored graph is called \emph{total rainbow} if its edges and internal vertices have distinct colors. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k \leq\ell$, the \emph{total rainbow $k$-connection number} of $G$, denoted by $trc_k(G)$, is the minimum number of colors used in a total coloring of $G$ to make $G$ \emph{total rainbow $k$-connected}, that is, any two vertices of $G$ are connected by $k$ internally vertex-disjoint total rainbow paths. In this paper, we study the computational complexity of total rainbow $k$-connection number of graphs. We show that it is NP-complete to decide whether $trc_k(G)=3$.