New results on word-representable graphs

TL;DR

This paper investigates the class of word-representable graphs, showing that not all graphs with maximum degree 4 are word-representable and deriving the asymptotic number of such graphs.

Contribution

It answers an open question by proving some degree-4 graphs are not word-representable and determines the asymptotic count of word-representable graphs.

Findings

Not all graphs with maximum degree 4 are word-representable.

The number of n-vertex word-representable graphs is approximately 2^{n^2/3} for large n.

The class of word-representable graphs includes several well-known graph families.

Abstract

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 $(x,y)\in E$ for each $x\neq y$. 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 [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 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 $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$.