EQUIMATCHABLE REGULAR GRAPHS

A graph in which all maximal matchings have the same size is called equimatch- able. We investigate equimatchable regular graphs and prove that when r is an odd positive integer, the only connected equimatchable r-regular graphs are K r+1 and K r,r . For an even r, we show that there are finitely many connected equimatchable r-regular graphs and when r = 4, we find all of them. Finally, a classification of all connected triangle-free equimatchable regular graphs is presented.