Tight bound for matching (vol 23, pg 322, 2012)

The bound on the size of maximum matching given in the paper as follows: M >= {m/2 - m/2L, if k = 2 m/k - m/(k + L)k, if k > 2 is not accurate. We now correct the bound to M >= m/k - m/Lk, where M is the size of the maximum matching in any graph of degree k, m edges and length of the smallest odd size cycle L. When no simple odd-length cycle exists it is known previously that M >= m/k. We show that the new bound given here is tight.