Packing and Covering Complete Bipartite Multidigraphs with 6-circuits

Abstract

Let C(subscript k) denote a circuit of length k. In a multidigraph G, a C(subscript k)-packing is a set P={G1, G2,…, G(subscript s)} of arc-disjoint subdigraphs of G such that each G(subscript i) is isomorphic to C(subscript k), and a C(subscript k)-packing is maximum if it has the maximum number of members among all packings; a C(subscript k)-covering is a set R={G1, G2,…, G(subscript t)} of subdigraphs of G such that each G(subscript i) is isomorphic to C(subscript k) and every arc of G appears in at least one member of R, and a C(subscript k) covering is minimum if it has the minimum number of members among all coverings. In this paper the problem for finding a maximum C4-packing and a minimum C4-covering of the complete bipartite multidigraph is solved.