On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions

TL;DR

This paper explores Hamilton decompositions of line graphs derived from non-Hamiltonian and special regular graphs, revealing new existence results and answering open questions in graph theory.

Contribution

It demonstrates the existence of non-Hamiltonian regular graphs with line graphs that have Hamilton decompositions, and constructs graphs without separating transitions whose line graphs lack such decompositions.

Findings

Existence of non-Hamiltonian k-regular graphs with Hamilton-decomposable line graphs for k≥4

Existence of k-regular graphs without separating transitions with non-Hamiltonian line graphs for k≥3

Answers to open questions about Hamilton decompositions in line graphs

Abstract

In contrast with Kotzig's result that the line graph of a $3$-regular graph $X$ is Hamilton decomposable if and only if $X$ is Hamiltonian, we show that for each integer $k\geq 4$ there exists a simple non-Hamiltonian $k$-regular graph whose line graph has a Hamilton decomposition. We also answer a question of Jackson by showing that for each integer $k\geq 3$ there exists a simple connected $k$-regular graph with no separating transitions whose line graph has no Hamilton decomposition.