Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the representation number of a crown graph   A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the
alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$
is an edge in $E$. It is known that any word-representable graph $G$ is
$k$-word-representable for some $k$, that is, there exists a word $w$
representing $G$ such that each letter occurs exactly $k$ times in $w$. The
minimum such $k$ is called $G$'s representation number.
  A crown graph $H_{n,n}$ is a graph obtained from the complete bipartite graph
$K_{n,n}$ by removing a perfect matching. In this paper we show that for $n\geq
5$, $H_{n,n}$'s representation number is $\lceil n/2 \rceil$. This result not
only provides a complete solution to the open Problem 7.4.2 in \cite{KL}, but
also gives a negative answer to the question raised in Problem 7.2.7 in
\cite{KL} on 3-word-representability of bipartite graphs. As a byproduct we
obtain a new example of a graph class with a high representation number.

16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat