On the number of edge-disjoint one factors and the existence of \(k\)-factors in complete multipartite graphs (Q1126293)

The authors investigate the maximum number of edge-disjoint 1-factors in \(K_{p_1,\dots,p_n}\), the complete multipartite graph with \(p_i\) vertices in the \(i\)th class. They settle this problems in the case where this number is greater than \(\delta(K_{p_1,\dots,p_n})-p_2\), where \(p_1\leq p_2\leq\cdots\leq p_n\), and \(\delta(G)\) is the minimum degree of graph \(G\). To do this they find necessary and sufficient conditions for the existence of a \(k\)-factor in \(K_{p_1,\dots,p_n}\).