Many-to-many two-disjoint path covers in cylindrical and toroidal grids

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex sets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. The many-to-many k-disjoint path cover is classified as paired if each source in S is further required to be paired with a specific sink in T , or unpaired otherwise. In this paper, we first establish a necessary and sufficient condition for a bipartite cylindrical grid to have a paired many-to-many 2-disjoint path cover joining S and T . Based on this characterization, we then prove that, provided the set S ∪ T contains the equal numbers of vertices from different parts of the bipartition, the bipartite cylindrical grid always has an unpaired many-to-many 2-disjoint path cover. Additionally, we show that such balanced vertex sets also guarantee the existence of a paired many-to-many 2-disjoint path cover for any bipartite toroidal grid even if an arbitrary edge is removed.