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 $λ$-number of $G$, denoted by $λ(G)$, 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 $Δ$, then $λ(G)\leq Δ+2$. Calamoneri and Petreschi proved that this conjecture is true when $Δ\geq 8$ but false when $Δ=3$. Meanwhile, they proved that $λ(G)\leq Δ+5$ for any outerplanar graph $G$ with $Δ=3$ and asked whether or not this bound is sharp. In this paper we answer this question by proving that $λ(G)\leq Δ+ 3$ for every outerplanar graph with maximum degree $Δ=3$. We also show that this bound $Δ+ 3$ can be achieved by infinitely many outerplanar graphs with $Δ=3$.

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