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 ( t + 1 ) k - 1 2 edges from G is not less than 1 2 ( n - t ) . In particular, taking t = 0 , we deduce that the resulting graph deleting any ⌊ k - 1 2 ⌋ 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 ≤ 2 k - 2 . Further, the resulting graph deleting any k + l - 1 2 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.