On sufficient degree conditions for a graph to be disjoint path coverable

A graph G is many-to-many t -disjoint path coverable if, for any two disjoint vertex subsets X = { x 1 , x 2 , … , x t } and Y = { y 1 , y 2 , … , y t } of V ( G ) , there exist t -disjoint paths such that each path connects x i and y τ ( i ) for some permutation τ of { 1 , 2 , … , t } and the union of these paths covers every vertex of G . By modifying the sets X and Y to X = { x } and Y = { y 1 , y 2 , … , y t } or X = { x } and Y = { y } , we can obtain the definitions of one-to-many t-disjoint path coverable and one-to-one t -disjoint path coverable , respectively. In this article, we obtain sufficient degree sum conditions on vertices at distance 2 for a graph to be one-to-one, one-to-many and many-to-many t -disjoint path coverable. Moreover, the bounds on the degree sums and the results obtained are all best possible. Our results also imply related results on Ore-type degree conditions.