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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 -connected graph G and an integer k with 1 ≤ k ≤ , 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.
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