Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Graphs of kei and their diameters   A kei on $[n]$ can be thought of as a set of maps $(f_x)_{x \in [n]}$, where
each $f_x$ is an involution on $[n]$ such that $(x)f_x = x$ for all $x$ and
$f_{(x)f_y} = f_yf_xf_y$ for all $x$ and $y$. We can think of kei as loopless,
edge-coloured multigraphs on $[n]$ where we have an edge of colour $y$ between
$x$ and $z$ if and only if $(x)f_y = z$; in this paper we show that any
component of diameter $d$ in such a graph must have at least $2^d$ vertices and
contain at least $2^{d-1}$ edges of the same colour. We also show that these
bounds are tight for each value of $d$.

17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
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
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics