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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 thatfor each integer $k\geq 4$ there exists a simple non-Hamiltonian $k$-regulargraph whose line graph has a Hamilton decomposition. We also answer a questionof Jackson by showing that for each integer $k\geq 3$ there exists a simpleconnected $k$-regular graph with no separating transitions whose line graph hasno Hamilton decomposition.

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