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A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertexset $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegativeintegers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edgeincident to $x$, and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are a pair of adjacentvertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$.The $(2,1)$-total labeling number $\lambda^T_2(G)$ of a graph $G$ is defined asthe minimum $k$ among all possible assignments. In [D. Chen and W. Wang.(2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585--2593(2007)], Chen and Wang conjectured that all outerplanar graphs $G$ satisfy  $\lambda^T_2(G) \leq \Delta(G)+2$, where $\Delta(G)$ is the maximum degree of$G$, while they also showed that it is true for $G$ with $\Delta(G)\geq 5$. Inthis paper, we solve their conjecture completely, by proving that  $\lambda^T_2(G) \leq \Delta(G)+2$ even in the case of $\Delta(G)\leq 4 $.

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