Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On generalized list $\G$-free colorings of graphs   For given graph $H$ and graphical property $P$, the conditional chromatic
number $\chi(H,P)$ of $H$, is the smallest number $k$, so that $V(H)$ can be
decomposed into sets $V_1,V_2,\ldots, V_k$, in which $H[V_i]$ satisfies the
property $P$, for each $1\leq i\leq k$. When property $P$ be that each color
class contains no copy of $G$, we write $\chi_{G}(H)$ instead of $\chi(G,P)$,
which is called the $G$-free chromatic number. Due to this, we say $H$ has a
$k$-$G$-free coloring if there is a map $c : V(H) \longrightarrow
\{1,\ldots,k\}$, so that each of the color classes of $c$ be $G$-free. Assume
that for each vertex $v$ of a graph $H$ is assigned a set $L(V)$ of colors,
called a color list. Set $g(L) = \{g(v): v\in V(H)\}$, that is the set of
colors chosen for the vertices of $H$ under $g$. An $L$-coloring $g$ is called
$G$-free, so that:
  \begin{itemize} \item $g(v)\in L(v)$, for any $v\in V(H)$. \item $ H[V_i]$ is
$G$-free for each $i=1,2,\ldots, L$. \end{itemize} If there exists an
$L$-coloring of $H$, then $H$ is called $L$-$G$-free-colorable. A graph $H$ is
said to be $k$-$G$-free-choosable if there exists an $L$-coloring for any
list-assignment $L$ satisfying $|L(V)|\geq k$ for each $v\in V(H)$, and
$H[V_i]$ be $G$-free for each $i=1,2,\ldots, L$. Let graph $H$ and a collection
of graphs $\G$ are given, the $\chi_{\G}^L(H)$ of $H$ is the last integer $k$,
so that $H$ is $k$-$\G$-free-choosable i.e. $H[V_i]$ is $\G$-free for each
$i=1,2,\ldots, k$ i.e. contains no copy of any member of $\G$. In this article,
we show that $\chi_G^L(H)=\chi_G(H)$ for some graph $H$ and $G$,
$\chi_G^L(H\oplus H')\leq \chi_G^L(H)+\chi_G^L(H')$ for each $G$, $H$, and
$H'$. Also, we show that $\chi_{\G}(H\oplus K_n)=\chi^L_{\G}(H\oplus K_n)$,
where $\G$ is a collection of all $d$-regular graphs, and some $n$.

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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an