On the representation number of a crown graph

  title={On the representation number of a crown graph},
  author={Marc Glen and Sergey Kitaev and Artem V. Pyatkin},
  journal={Discret. Appl. Math.},
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… Expand
The k-cube is k-representable.
A graph is called $k$-representable if there exists a word $w$ over the nodes of the graph, each node occurring exactly $k$ times, such that there is an edge between two nodes $x,y$ if and only afterExpand
On the Representation Number of Bipartite Graphs
A polynomial time relabeling algorithm is proposed to produce a word representing a given bipartite graph which is a concatenation of permutations of the graph’s vertices, which gives an upper bound for the representation number of bipartites. Expand
Polygon-circle and word-representable graphs
The relationship between the independently-studied polygon-circle graphs and word-representable graphs is described and it is shown that neither of these two classes is included in the other one by showing that the word- Representable Petersen graph and crown graphs are notpolygon- circle, while the non-word-Representable wheel graph W5 is polygon -circle. Expand
S ep 2 02 1 On the Representation Number of Bipartite Graphs
A word-representable graph is a simple graph G which can be represented by a word w over the vertices of G such that any two vertices are adjacent in G if and only if they alternate in w. It is knownExpand
Symbolic Powers of Certain Cover Ideals of Graphs
In this paper, we compute the regularity and Hilbert series of symbolic powers of cover ideal of a graph G when G is either a crown graph or a complete multipartite graph. We also compute theExpand


On Graphs with Representation Number 3
  • S. Kitaev
  • Mathematics, Computer Science
  • J. Autom. Lang. Comb.
  • 2013
It is shown that any prism belongs to the class of graphs with representation number 3, denoted by $\mathcal{R}_3$, and that two particular operations of extending graphs preserve the property of being in this class. Expand
Word Problem of the Perkins Semigroup via Directed Acyclic Graphs
It is shown here that the computational complexity problem that has as instance {u,v} where u,v are words of finite length, and question “Is Alt(u) = Alt(v)?”, is co-NP-complete. Expand
Semi-transitive orientations and word-representable graphs
An effective characterization of word-representable graphs in terms of orientations is given, showing that the recognition problem is in NP, and that word- Representable graphs include all 3-colorable graphs. Expand
On Representable Graphs
It is proved that a graph is representable if and only if it is k-representable for some k, and some wide classes of graphs are proven to be 2- and 3- representable. Expand
The Maximum Clique Problem
This paper considers the problem of finding the largest clique in a graph using the notation ES to represent the subset of edges which have both endpoints in clique S, and concludes that the induced graph GS = ( S, ES ) is complete. Expand
The Perkins Semigroup has Co-Np-Complete Term-Equivalence Problem
  • S. Seif
  • Mathematics, Computer Science
  • Int. J. Algebra Comput.
  • 2005
It is proved here that $\mathbf{B^1_2}$, the six-element Perkins semigroup, has co-NP-complete term-equivalence problem, a result which leads to the completion of the classification of he term-Equivalence problems for monoid extensions of aperiodic Rees matrix semigroups. Expand
Coloring circle graphs
  • Jakub Cerný
  • Mathematics, Computer Science
  • Electron. Notes Discret. Math.
  • 2007
The bound χ ( G ) ⩽ ω ⋅ log n is presented which shows, that the circle graphs with small maximum clique and large chromatic number must have many vertices. Expand
Words and Graphs
  • S. Kitaev, V. Lozin
  • Computer Science
  • Monographs in Theoretical Computer Science. An EATCS Series
  • 2015
This is the first comprehensive introduction to the theory of word-representable graphs, a generalization of several classical classes of graphs, and a new topic in discrete mathematics. AfterExpand
3-Coloring in Time O(1.3289^n)
This work considers worst case time bounds for several NP-complete problems, based on a constraint satisfaction (CSP) formulation of these problems, and shows that n-variable (3, 2)-CSP instances can be solved in time O(1.3645n). Expand
ω-Perfect graphs
Abstractω-Perfect graph is defined and some classes of ω-perfect graphs are described, although the characterization of the complete class of ω-perfect graphs remains an open question. A bound on theExpand