For positive integers d and m, let PdÅ,m(G) denote the property that between each pair of vertices of the graph G, there are m internally disjoint paths of length at most d. For a positive integer t, a graph G satisfies the minimum generalized degree condition δt(G)≥s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that insure that PdÅ,m(G) is satisfied are investigated. For example, if for fixed positive integers t≥5, d≥5t2, and m≥2, an m-connected graph G of order n satisfies the generalized degree condtion δt(G)≥(t/(t+1))(5n/(d+2))+(m-1)d+3t2, then for n sufficiently large G has property PdÅ,m(G). Also, if the order of magnitude of δt(G) is decreased, then PdÅ,m(G) will not hold; so the result is sharp in terms of order of magnitude of δt(G).
