Factorizations of 4-Regular Graphs and Petersen′s Theorem

On the basis of the observation that a 3-regular graph has a perfect matching if and only if its line graph has a triangle-free 2-factorisation, we show that a connected 4-regular graph has a triangle-free 2-factorisation, provided it has no more than two cut-vertices belonging to a triangle. This result is equivalent to Petersen′s theorem about the existence of perfect matchings in 3-regular graphs.