TY - JOUR

T1 - New results on word-representable graphs

AU - Collins, Andrew

AU - Kitaev, Sergey

AU - Lozin, Vadim V.

N1 - “NOTICE: this is the author’s version of a work that was accepted for publication in Discrete Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete Applied Mathematics, [VOL 216, Part 1, (20/11/14)] DOI: 10.1016/j.dam.2014.10.024¨

PY - 2017/1/10

Y1 - 2017/1/10

N2 - A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if (x,y)∈E(x,y)∈E for each x≠yx≠y. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from Halldórsson et al. (2011), in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of nn-vertex word-representable graphs is View the MathML source2n23+o(n2).

AB - A graph G=(V,E)G=(V,E) is word-representable if there exists a word ww over the alphabet VV such that letters xx and yy alternate in ww if and only if (x,y)∈E(x,y)∈E for each x≠yx≠y. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from Halldórsson et al. (2011), in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of nn-vertex word-representable graphs is View the MathML source2n23+o(n2).

KW - semi-transitive orientation

KW - speed of hereditary properties

KW - hereditary property of graphs

U2 - 10.1016/j.dam.2014.10.024

DO - 10.1016/j.dam.2014.10.024

M3 - Article

VL - 216

SP - 136

EP - 141

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - Part 1

ER -