Minimum degree of bipartite graphs and the existence ofk-factors

Abstract

LetG be a bipartite graph with bipartition (X, Y) andk a positive integer. If

$$\left| X \right| = \left| Y \right|,$$

((i))

$$\delta (G) \geqslant \left\lceil {\frac{{\left| X \right|}}{2}} \right\rceil \geqslant k,$$

((ii))

$\left| X \right| \geqslant 4k - 4\sqrt k + 1$ when |X| is odd and |X| ≥ 4k − 2 when |X| is even, thenG has ak-factor.