Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Complete and almost complete minors in double-critical 8-chromatic
  graphs   A connected $k$-chromatic graph $G$ is said to be {\it double-critical} if
for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A
longstanding conjecture of Erd\H{o}s and Lov\'asz states that the complete
graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft
[\it{Electron. J. Combin.}, 17(1): Research Paper 87, 2010] proved that every
double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$ minor. It
remains unknown whether an arbitrary double-critical $8$-chromatic graph
contains a $K_8$ minor, but in this paper we prove that any double-critical
$8$-chromatic contains a $K_8^-$ minor; here $K_8^-$ denotes the complete
$8$-graph with one edge missing. In addition, we observe that any
double-critical $8$-chromatic graph with minimum degree different from $10$ and
$11$ contains a $K_8$ minor.

12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat