Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Iteration Number of Colour Refinement   The Colour Refinement procedure and its generalisation to higher dimensions,
the Weisfeiler-Leman algorithm, are central subroutines in approaches to the
graph isomorphism problem. In an iterative fashion, Colour Refinement computes
a colouring of the vertices of its input graph.
  A trivial upper bound on the iteration number of Colour Refinement on graphs
of order n is n-1. We show that this bound is tight. More precisely, we prove
via explicit constructions that there are infinitely many graphs G on which
Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite
families that we present, we show that for every natural number n >= 10, there
are graphs on n vertices on which Colour Refinement requires at least n-2
iterations to reach stabilisation.

7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th