Labeling outerplanar graphs with maximum degree three

An $L(2, 1)$-labeling of a graph $G$ is an assignment of a nonnegative integer to each vertex of $G$ such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span of such a labeling is the difference between the largest and smallest integers used. The $\lambda$-number of $G$, denoted by $\lambda(G)$, is the minimum span over all $L(2, 1)$-labelings of $G$. Bodlaender {\it et al.} conjectured that if $G$ is an outerplanar graph of maximum degree $\Delta$, then $\lambda(G)\leq \Delta+2$. Calamoneri and Petreschi proved that this conjecture is true when $\Delta \geq 8$ but false when $\Delta=3$. Meanwhile, they proved that $\lambda(G)\leq \Delta+5$ for any outerplanar graph $G$ with $\Delta=3$ and asked whether or not this bound is sharp. In this paper we answer this question by proving that $\lambda(G)\leq \Delta + 3$ for every outerplanar graph with maximum degree $\Delta=3$. We also show that this bound $\Delta + 3$ can be achieved by infinitely many outerplanar graphs with $\Delta=3$.