Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note On Vertex Distinguishing Edge colorings of Trees   A proper edge coloring of a simple graph $G$ is called a vertex
distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and
$v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs
from the set of the colors assigned to the edges incident to $v$. The minimum
number of colors required for all vdecs of $G$ is denoted by $\chi\,'_s(G)$
called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of
vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with
$n_2(T)\leq n_1(T)$ holds $\chi\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or
one of two particular trees with $D(T) =4$, and $\chi\,'_s(T)=n_1(T)$
otherwise; furthermore $\chi\,'_{es}(T)=\chi\,'_s(T)$ when $|E(T)|\leq
2(n_1(T)+1)$, where $\chi\,'_{es}(T)$ is the equitable vdec chromatic number of
$T$.

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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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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: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph