The (2,1)-Total Labeling Number of Outerplanar Graphs Is at Most Δ + 2

Abstract

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 $\lambda^T_2(G)$ of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy $\lambda^T_2(G) \leq \Delta(G)+2$, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that $\lambda^T_2(G) \leq \Delta(G)+2$ even in the case of Δ(G) ≤ 4 .