Labeling outerplanar graphs with maximum degree three

An L(2,1)-labeling of a graph G is an assignment of a non-negative 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 @l-number of G, denoted by @l(G), is the minimum span over all L(2,1)-labelings of G. Bodlaender et al. conjectured that if G is an outerplanar graph of maximum degree @D, then @l(G)@[email protected]+2. Calamoneri and Petreschi proved that this conjecture is true when @D>=8 but false when @D=3. Meanwhile, they proved that @l(G)@[email protected]+5 for any outerplanar graph G with @D=3 and asked whether or not this bound is sharp. In this paper we answer this question by proving that @l(G)@[email protected]+3 for every outerplanar graph with maximum degree @D=3. We also show that this bound @D+3 can be achieved by infinitely many outerplanar graphs with @D=3.