# On the representation number of a crown graph

Marc Glen, Sergey Kitaev, Artem Pyatkin

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number.
A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number.
Original language English 89-93 5 Discrete Applied Mathematics 244 23 Mar 2018 https://doi.org/10.1016/j.dam.2018.03.013 Published - 31 Jul 2018

### Fingerprint

Byproducts
Graph in graph theory
Representability
Graph Classes
Complete Bipartite Graph
Perfect Matching
Bipartite Graph
Alternate
Open Problems
If and only if

### Keywords

• word-representable graph
• crown graph
• cocktail party graph
• representation number

### Cite this

@article{bd457043a16c4179b27cdfbdd56684b2,
title = "On the representation number of a crown graph",
abstract = "A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number.A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number.",
keywords = "word-representable graph, crown graph, cocktail party graph, representation number",
author = "Marc Glen and Sergey Kitaev and Artem Pyatkin",
year = "2018",
month = "7",
day = "31",
doi = "10.1016/j.dam.2018.03.013",
language = "English",
volume = "244",
pages = "89--93",
journal = "Discrete Applied Mathematics",
issn = "0166-218X",

}

On the representation number of a crown graph. / Glen, Marc; Kitaev, Sergey; Pyatkin, Artem.

In: Discrete Applied Mathematics, Vol. 244, 31.07.2018, p. 89-93.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On the representation number of a crown graph

AU - Glen, Marc

AU - Kitaev, Sergey

AU - Pyatkin, Artem

PY - 2018/7/31

Y1 - 2018/7/31

N2 - A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number.A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number.

AB - A graph G = (V,E) is word-representable if there exists a word ω over the alphabet V such that letters x and y alternate in ω if and only if xy is an edge in E . It is known (Kitaev and Pyatkin, 2008) that any word-representable graph G is k-word-representable for some k, that is, there exists a word ω representing G such that each letter occurs exactly k times in ω. The minimum such k is called G’s representation number.A crown graph (also known as a cocktail party graph) Hn,n is a graph obtained from the complete bipartite graph Kn,n by removing a perfect matching. In this paper, we show that for n≥ 5,Hn,n ’s representation number is [n / 2]. This result not only provides a complete solution to the open Problem 7.4.2 in Kitaev and Lozin (2015), but also gives a negative answer to the question raised in Problem 7.2.7 in Kitaev and Lozin (2015) on 3-word-representability of bipartite graphs. As a byproduct, we obtain a new example of a graph class with a high representation number.

KW - word-representable graph

KW - crown graph

KW - cocktail party graph

KW - representation number

UR - https://www.sciencedirect.com/journal/discrete-applied-mathematics

U2 - 10.1016/j.dam.2018.03.013

DO - 10.1016/j.dam.2018.03.013

M3 - Article

VL - 244

SP - 89

EP - 93

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -