Abstract
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 |
|---|---|
| Pages (from-to) | 89-93 |
| Number of pages | 5 |
| Journal | Discrete Applied Mathematics |
| Volume | 244 |
| Early online date | 23 Mar 2018 |
| DOIs | |
| Publication status | Published - 31 Jul 2018 |
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Keywords
- word-representable graph
- crown graph
- cocktail party graph
- representation number
Cite this
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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 journal › Article
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 -