Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An improved procedure for colouring graphs of bounded local density   We develop an improved bound for the chromatic number of graphs of maximum
degree $\Delta$ under the assumption that the number of edges spanning any
neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed
$0<\sigma<1$. The leading term in the reduction of colours achieved through
this bound is best possible as $\sigma\to0$. As two consequences, we advance
the state of the art in two longstanding and well-studied graph colouring
conjectures, the Erd\H{o}s-Ne\v{s}et\v{r}il conjecture and Reed's conjecture.
We prove that the strong chromatic index is at most $1.772\Delta^2$ for any
graph $G$ with sufficiently large maximum degree $\Delta$. We prove that the
chromatic number is at most $\lceil 0.881(\Delta+1)+0.119\omega\rceil$ for any
graph $G$ with clique number $\omega$ and sufficiently large maximum degree
$\Delta$. Additionally, we show how our methods can be adapted under the
additional assumption that the codegree is at most $(1-\sigma)\Delta$, and
establish what may be considered first progress towards a conjecture of Vu.

11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int