Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Chromatic number, clique subdivisions, and the conjectures of Haj\'os
  and Erd\H{o}s-Fajtlowicz   For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$
denote the order of the largest clique subdivision in $G$. Let H(n) be the
maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous
conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every
graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This
conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further
showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some
absolute constant $c>0$. In 1981 they conjectured that this bound is tight up
to a constant factor in that there is some absolute constant $C$ such that
$\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this
paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our
proof, which might be of independent interest, is an estimate on the order of
the largest clique subdivision which one can find in every graph on $n$
vertices with independence number $\alpha$.

28
The categories of this paper are: alg-geom
1
The categories of this paper are: cs
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci