New results on word-representable graphs

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• 作者：Andrew Collins^a ; ^{A.Collins.2@warwick.ac.uk} ; Sergey Kitaev^b ; ^{sergey.kitaev@cis.strath.ac.uk} ; Vadim V. Lozin^a ; ^c ; ^{V.Lozin@warwick.ac.uk}
• 关键词：Semi-transitive orientation ; Hereditary property of graphs ; Speed of hereditary properties
• 刊名：Discrete Applied Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：216
• 期：part_P1
• 页码：136-141
• 全文大小：402 K
• 卷排序：216

文摘

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 2n23+o(n2).