The rainbow connectivity of a graph

A path P in an edge‐colored graph (not necessarily a proper edge‐coloring) is a rainbow path if no two edges of P are colored the same. For an ℓ‐connected graph G and an integer k with 1 ≤ k ≤ ℓ, the rainbow k ‐connectivity rc k ( G ) of G is the minimum integer j for which there exists a j ‐edge‐coloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. The rainbow k ‐connectivity of the complete graph K n is studied for various pairs k , n of integers. It is shown that for every integer k ≥ 2, there exists an integer f ( k ) such that rc k ( K n ) = 2 for every integer n ≥ f ( k ). We also investigate the rainbow k ‐connectivity of r ‐regular complete bipartite graphs for some pairs k , r of integers with 2 ≤ k ≤ r . It is shown that for each integer k ≥ 2, there exists an integer r such that rc k ( K r , r ) = 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009