Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complexity of the conditional colorability of graphs   For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a
proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree
$d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different
colors. The smallest integer $k$ for which a graph $G$ has a conditional
$(k,r)$-coloring is called the $r$th order conditional chromatic number,
denoted by $\chi_r(G)$. It is easy to see that the conditional coloring is a
generalization of the traditional vertex coloring for which $r=1$. In this
paper, we consider the complexity of the conditional colorings of graphs. The
main result is that the conditional $(3,2)$-colorability is $NP$-complete for
triangle-free graphs with maximum degree at most 3, which is different from the
old result that the traditional 3-colorability is polynomial solvable for
graphs with maximum degree at most 3. This also implies that it is
$NP$-complete to determine if a graph of maximum degree 3 is $(3,2)$- or
$(4,2)$-colorable. Also we have proved that some old complexity results for
traditional colorings still hold for the conditional colorings.

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The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
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The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
19
The categories of this paper are: eess