Fractional matchings on regular graphs

Every k-regular graph has a fractional perfect matching via assigning each edge a fractional number 1/k. How many edges are deleted from a regular graph so that the resulting graph still has a fractional perfect matching? Let G be a k-regular graph with n vertices. In this paper, we prove that the fractional matching number of the resulting graph deleting any (sic)(t+1)k-1/2(sic) edges from G is not less than 1/2(n - t). In particular, taking t = 0, we deduce that the resulting graph deleting any (sic)k-1/2(sic) edges from G has a fractional perfect matching. Specially, we can delete any k - 1 edges from G other than exceptions such that the resulting graph has a fractional perfect matching when n <= 2k - 2. Further, the resulting graph deleting any (sic)k+l-1/2(sic) edges from a k-regular l-edge-connected graph with an even number of vertices has a fractional perfect matching. As applications, some values or bounds on the fractional matching preclusion number of regular graphs are deduced immediately.