As the title says, let $k \geq 2$ be a positive integer and let $G$ be a $(k-1)$-edge-connected $k$-regular graph with an even number of vertices. Then, for every edge $e$ of the graph there is a perfect matching of $G$ containing $e$. 


First, I was wondering if this is new and if there are approaches different from mine (and for that reason I won't mention my idea at this point)


Some background: The particular case k=3 is a theorem of Petersen; I had it as a graph theory exam problem last year; my solution led to this generalization.



A k-1 edge connected k regular graph is matching covered
If you look at the math review of the following:


MR0317999 (47 #6548) 
Plesník, Ján
Connectivity of regular graphs and the existence of 1-factors. 
Mat. Časopis Sloven. Akad. Vied 22 (1972), 310–318. 


You will see that this result was already known in 1972, and if you look at the actual paper (available for free on the Czech digital text collection), the author attributes it to C. Berge, proved in his graph theory book (1958) (where this is the content of theorem 6 and 7 of Chapter 18).


So, alas, this is not new.


