Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On color-critical ($P_{5},\overline{P}_5$)-free graphs   A graph is $k$-critical if it is $k$-chromatic but each of its proper induced
subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical
$P_5$-free graphs is finite, but there is an infinite number of $k$-critical
$P_5$-free graphs for each $k \geq 5$. We show that the number of $k$-critical
$(P_5, \overline{P}_5)$-free graphs is finite for every fixed $k$. Our result
implies the existence of a certifying algorithm for $k$-coloring $(P_5,
\overline{P}_5)$-free graphs.

16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin