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

Abstract

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