On the Existence of Word-representable Line Graphs of Non-word-representable Graphs

TL;DR

This paper investigates whether line graphs of non-word-representable graphs can be word-representable, providing computational evidence that they can be, thus challenging previous assumptions in graph theory.

Contribution

It formulates an optimization problem for 3-semi-transitive graphs and demonstrates through computational experiments that line graphs of non-word-representable graphs can be word-representable.

Findings

Line graphs of non-word-representable graphs can be word-representable.

The optimization problem can identify 3-semi-transitive line graphs.

Computational experiments support the existence of such graphs.

Abstract

An open question in the theory of word-representable graphs for the past decade has been whether the line graph of a non-word-representable graph is always non-word-representable. By formulating an appropriate optimization problem for the decision problem of 3-semi-transitive graphs, we show that the line graph of a non-word-representable graph can be word-representable. Using IBM's CPLEX solver, we demonstrate for several known word-representable and non-word-representable graphs that the line graph of a graph is 3-semi-transitive when there is a solution to the optimization problem. This results in an example where the line graph of a non-word-representable graph is both 3-semi-transitive and semi-transitive and thus is word-representable.