Existence of $u$-representation of graphs

TL;DR

This paper investigates $u$-representable graphs, revealing that for words $u$ of length at least 3, all graphs are $u$-representable, thus highlighting only two meaningful cases: $u=11$ and $u=12$.

Contribution

It proves that for any $u$ of length at least 3, every graph is $u$-representable, simplifying the classification of $u$-representable graphs.

Findings

All graphs are $u$-representable for $u$ of length ≥ 3.

The only non-trivial cases are $u=11$ and $u=12$.

The class of $u$-representable graphs is trivial for longer words.

Abstract

Recently, Jones et al. introduced the study of $u$-representable graphs, where $u$ is a word over $\{1,2\}$ containing at least one 1. The notion of a $u$-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is $11\cdots 1$-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of $u$-representable graphs at least for small values of $u$. In this paper we show that if $u$ is of length at least 3 then any graph is $u$-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of $u$-representable graphs, namely, those of $u=11$ and $u=12$.