Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Harmonious Coloring of Trees with Large Maximum Degree   A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that
every pair of colors appears on at most one pair of adjacent vertices. The
harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors
needed for a harmonious coloring of $G$. We show that if $T$ is a forest of
order $n$ with maximum degree $\Delta(T)\geq \frac{n+2}{3}$, then $$h(T)=
  \Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$;
  \Delta(T)+1, & otherwise.
  $$ Moreover, the proof yields a polynomial-time algorithm for an optimal
harmonious coloring of such a forest.

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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
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The categories of this paper are: stat
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: physics
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The categories of this paper are: dg-ga