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 λ2 (G) of a graph G is defined as the minimum k among all possible assignments. In [4], Chen and Wang conjectured that all outerplanar graphs G satisfy λ2 (G) ≤ ∆(G) + 2, where ∆(G) is the maximum degree of G, while they also showed that it is true for G with ∆(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2 (G) ≤ ∆(G) + 2 even in the case of ∆(G) ≤ 4.