Hamilton decompositions of line graphs

TL;DR

This paper proves conditions under which line graphs of certain regular graphs are Hamilton decomposable, extending existing results and confirming Bermond's conjecture about Hamilton decomposability of line graphs.

Contribution

It establishes new sufficient conditions for the Hamilton decomposability of line graphs of regular graphs, extending prior results and confirming Bermond's conjecture.

Findings

Line graphs of regular graphs with even degree and a Hamilton cycle are Hamilton decomposable.

Line graphs of regular graphs with odd degree and a Hamiltonian 3-factor are Hamilton decomposable.

The results extend Kotzig's theorem and prove Bermond's conjecture.

Abstract

It is proved that if a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor, then its line graph is Hamilton decomposable. This result partially extends Kotzig's result that a $3$-regular graph is Hamiltonian if and only if its line graph is Hamilton decomposable, and proves the conjecture of Bermond that the line graph of a Hamilton decomposable graph is Hamilton decomposable.