Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hadwiger's conjecture for graphs with forbidden holes   Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the
largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor.
A hole in $G$ is an induced cycle of length at least four. Hadwiger's
Conjecture from 1943 states that for every graph $G$, $h(G)\ge \chi(G)$, where
$\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more
evidence for Hadwiger's conjecture by showing that if a graph $G$ with
independence number $\alpha(G)\ge3$ has no hole of length between $4$ and
$2\alpha(G)-1$, then $h(G)\ge\chi(G)$. We also prove that if a graph $G$ with
independence number $\alpha(G)\ge2$ has no hole of length between $4$ and
$2\alpha(G)$, then $G$ contains an odd clique minor of size $\chi(G)$, that is,
such a graph $G$ satisfies the odd Hadwiger's conjecture.

17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas