Line Graphs of Non-Word-Representable Graphs are Not Always Non-Word-Representable

Abstract

A graph is said to be word-representable if there exists a word over its vertex set such that any two vertices are adjacent if and only if they alternate in the word. If no such word exists, the graph is non-word-representable. In the literature, there are examples of non-word-representable graphs whose line graphs are non-word-representable. However, it is an open problem to determine whether the line graph of a non-word-representable graph is always non-word-representable or not? In this work, we address the open problem by considering a class of non-word-representable graphs, viz., Mycielski graphs of odd cycles of length at least five, and show that their line graphs are word-representable.