A tight upper bound on the (2,12,1)-total labeling number of outerplanar graphs

A (2,1)(2,1)-total labeling of a graph G is an assignment f   from the vertex set V(G)V(G) and the edge set E(G)E(G) to the set {0,1,…,k}{0,1,…,k} of nonnegative integers such that |f(x)−f(y)|⩾2|f(x)−f(y)|⩾2 if x is a vertex and y is an edge incident to x  , and |f(x)−f(y)|⩾1|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)V(G)∪E(G). The (2,1)(2,1)-total labeling number λ2T(G) of G is defined as the minimum k   among all possible (2,1)(2,1)-total labelings of G. In 2007, Chen and Wang conjectured that all outerplanar graphs G   satisfy λ2T(G)⩽Δ(G)+2, where Δ(G)Δ(G) is the maximum degree of G. They also showed that it is true for G   with Δ(G)⩾5Δ(G)⩾5. In this paper, we solve their conjecture, by proving that λ2T(G)⩽Δ(G)+2, even when Δ(G)⩽4Δ(G)⩽4.