The recursive bound for tone chromatic numbers of subcubic outerplanar graphs

Let $G$ be a subcubic outerplanar graph, meaning that $G$ is outerplanar and has maximum degree at most $3$. For each positive integer $t$, let $\tau_t(G)$ denote the smallest number of colors in a $t$-tone coloring of $G$. The recursive bound. For every integer $t\ge 3$,

$$\tau_t(G)\le \tau_{t-1}(G)+t+1.$$

This is proposed as a relationship between successive tone chromatic numbers after the paper establishes the sharp bound $\tau_3(G)\le 11$ for subcubic outerplanar graphs. Its resolution is not specified in the supplied text.