Maximum Matching in Regular and Almost Regular Graphs

We present an O(n ²logn)‐time algorithm that finds a maximum matching in a regular graph with n vertices. More generally, the algorithm runs in O(rn ²logn) time if the difference between the maximum degree and the minimum degree is less than r. This running time is faster than applying the fastest known general matching algorithm that runs in $O(\sqrt{n}m)$ ‐time for graphs with m edges, whenever m=ω(rn ^1.5logn).