Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge
incident to $x$, and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are a pair of adjacent
vertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$.
The $(2,1)$-total labeling number $\lambda^T_2(G)$ of a graph $G$ 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, 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 $\Delta(G)\geq 5$. In
this 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 $.

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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg