A tight upper bound on the (2, 1)-total labeling number of outerplanar graphs

A (2,1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0,1,...,k} of nonnegative integers such that |f(x)-f(y)|≥2 if x is a vertex and y is an edge incident to x, and |f(x)-f(y)|≥1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2,1)-total labeling number λ _T ² (G) of G is defined as the minimum k among all possible (2,1)-total labelings of G. In 2007, Chen and Wang conjectured that all outerplanar graphs G satisfy λ _T ² (G)≤Δ(G)+2, where Δ(G) is the maximum degree of G. They also showed that it is true for G with Δ(G)≥5. In this paper, we solve their conjecture, by proving that λ _T ² (G)≤Δ(G)+2, even when Δ(G)≤4.