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 et al. conjectured that if G is an outerplanar graph of maximum degree ∆, then λ(G) ≤ ∆ + 2. Calamoneri and Petreschi proved that this conjecture is true when ∆ ≥ 8 but false when ∆ = 3. Meanwhile, they proved that λ(G) ≤ ∆ + 5 for any outerplanar graph G with ∆ = 3 and asked whether or not this bound is sharp. In this paper we answer this question by proving that λ(G) ≤ ∆ + 3 for every outerplanar graph with maximum degree ∆ = 3. We also show that this bound ∆ + 3 can be achieved by infinitely many outerplanar graphs with ∆ = 3