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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 $\\lambda(G)\\leq \\Delta+2$. 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