Geodesic
Stronger bounds for generalized degrees and Menger path systems
Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 2, pp. 167-177.

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For positive integers d and m, let P_d,m(G) denote the property that between each pair of vertices of the graph G, there are m internally vertex 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 ensure that P_d,m(G) is satisfied have been investigated. In particular, it has been shown, for fixed positive integers t ≥ 5, d ≥ 5t², and m, that if an m-connected graph G of order n satisfies the generalized degree condition δₜ(G) > (t/(t+1))(5n/(d+2))+(m-1)d+3t², then for n sufficiently large G has property P_d,m(G). In this note, this result will be improved by obtaining corresponding results on property P_d,m(G) using a generalized degree condition δₜ(G), except that the restriction d ≥ 5t² will be replaced by the weaker restriction d ≥ max5t+28,t+77. Also, it will be shown, just as in the original result, that if the order of magnitude of δₜ(G) is decreased, then P_d,m(G) will not, in general, hold; so the result is sharp in terms of the order of magnitude of δₜ(G).
Keywords: generalized degree, Menger 
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Faudree, R.; Tuza, Zs. Stronger bounds for generalized degrees and Menger path systems. Discussiones Mathematicae. Graph Theory, Tome 15 (1995) no. 2, pp. 167-177. http://geodesic.mathdoc.fr/item/DMGT_1995_15_2_a3/

[1] [CL] G. Chartrand and L. Lesniak, Graphs and Digraphs (Prindle Weber Schmidt Boston 1986).

[2] [FGL] R.J. Faudree, R.J. Gould and L. Lesniak, Generalized Degrees and Menger Path Systems, Discrete Applied Math. 37-38 (1992) 179-191, doi: 10.1016/0166-218X(92)90132-T.

[3] [FGS] R.J. Faudree, R.J. Gould and R.H. Schelp, Menger Path Systems, J. Combin. Math. Combin. Comp. 6 (1989) 9-21.

[4] [FJOST] R.J. Faudree, M.S. Jacobson, E.T. Ordman, R.H. Schelp and Zs. Tuza, Menger's Theorem and Short Paths, J. Combin. Math. Combin. Comp. 2 (1987) 235-253.

[5] [O] E.T. Ordman, Fault-tolerant Networks and Graph Connectivity, J. Combin. Math. Combin. Comp. 1 (1987) 191-205.