The rainbow connectivity of a graph

  • Authors:

  • Gary Chartrand;Garry L. Johns;Kathleen A. McKeon;Ping Zhang

  • Affiliations:

  • Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008;Department of Mathematical Sciences, Saginaw Valley State University, University Center, Michigan 48710-0001;Department of Mathematics, Connecticut College, New London, Connecticut 06320;Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008

  • Venue:
  • Networks
  • Year:
  • 2009

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Abstract

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 rck(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 Kn 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 rck(Kn) = 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 rck(Kr,r) = 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009