Fractional matchings on regular graphs

Abstract

Every k-regular graph has a fractional perfect matching via assigning each edge a fractional number $\frac{1}{k}$. 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 $\left\lfloor \frac{(t+1)k-1}{2}\right\rfloor$ edges from G is not less than $\frac{1}{2}(n-t)$. In particular, taking $t=0$, we deduce that the resulting graph deleting any $\lfloor \frac{k-1}{2}\rfloor$ 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 $n\le 2k-2$. Further, the resulting graph deleting any $\left\lfloor \frac{k+l-1}{2}\right\rfloor$ 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.