On the representation number of a crown graph

TL;DR

This paper determines the exact representation number of crown graphs, showing it is eil n/2 eil for n, solving an open problem and providing insights into word-representability of bipartite graphs.

Contribution

It establishes the precise representation number of crown graphs for all n, resolving an open problem and addressing a question about 3-word-representability of bipartite graphs.

Findings

Representation number of crown graphs is eil n/2 eil for n.

Provides a negative answer to the 3-word-representability question for bipartite graphs.

Introduces a new graph class with high representation number.

Abstract

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.