Changes
If the class sizes are ''almost equal'' (that is <math>|n_i-n_j|\le 1</math> for every <math>1\le i<j\le k</math>) then the answer to the above question is always YES by
the skew-supermodular variant<ref>A. Bernáth, T. Király, ''Covering skew-supermodular functions by hypergraphs of minimum total size'', Operations Research Letters 37 (2009), 345-350. [http:name="BeKi"//dx.doi.org/10.1016/j.orl.2009.04.002 DOI link]. [http://www.cs.elte.hu/egres/tr/egres-08-05.pdf EGRES Technical Report].</ref> of Schrijver's [[Supermodular colouring theorem]]. A special case of the problem is the following: given a bipartite graph ''G=(A,B,E)'' with <math> d_G(v)\le k</math> for every <math>v\in A\cup B</math>, and ''k'' nonnegative integers <math>n_1,n_2,\dots,n_k</math> satisfying <math>n_1+n_2+\dots+n_k = |E|</math>. Does there exist ''k'' disjoint matchings <math>M_1,M_2,\dots,M_k\subseteq E</math> such that <math>|M_i| = n_i</math> for every ''i=1,...,k''?
==References==
<references><ref name="BeKi">A. Bernáth, T. Király, ''Covering skew-supermodular functions by hypergraphs of minimum total size'', Operations Research Letters 37 (2009), 345-350. [http://dx.doi.org/10.1016/j.orl.2009.04.002 DOI link], [http://www.cs.elte.hu/egres/tr/egres-08-05.pdf EGRES Technical Report].</ref></references>
[[Category:Colouring]]
[[Category:Sub- and supermodular functions]]
[[Category:Open Problems]]
