Ore-type degree conditions for disjoint path covers in simple graphs

A many-to-many  kk-disjoint path cover   of a graph joining two disjoint vertex subsets SS and TT of equal size kk is a set of kk vertex-disjoint paths between SS and TT that altogether cover every vertex of the graph. It is classified as paired   if each source in SS is required to be paired with a specific sink in TT, or unpaired   otherwise. In this paper, we develop Ore-type sufficient conditions for the existence of many-to-many kk-disjoint path covers joining arbitrary vertex subsets SS and TT. Also, an Ore-type degree condition is established for the one-to-many kk-disjoint path cover, a variant derived by allowing to share a single source. The bounds on the degree sum are all best possible.