On the Representability of Line Graphs

TL;DR

This paper investigates the conditions under which line graphs are representable, proving that many common line graphs, including those of wheels and complete graphs, are non-representable, thus answering open questions in graph theory.

Contribution

It establishes new non-representability results for line graphs of wheels and complete graphs, and characterizes when iterated line graphs are non-representable.

Findings

Line graphs of n-wheels for n > 3 are non-representable.

Line graphs of complete graphs for n > 4 are non-representable.

Iterated line graphs of certain graphs are non-representable for k > 3.

Abstract

A graph G=(V,E) is 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) is in E for each x not equal to y. The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for n greater than 3, the line graph of an n-wheel is non-representable. This not only provides a new construction of non-representable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for n greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3.