Abstract
An edge-colored graph G is rainbow connected if every two vertices of G are connected by a path whose edges have distinct colors. The rainbow connection number of G, denoted by rc(G), is the minimum number of colors that are needed to make G rainbow connected. In this paper we give a Nordhaus–Gaddum-type result for the rainbow connection number. We prove that if G and \({\overline{G}}\) are both connected, then \({4\leq rc(G)+rc(\overline{G})\leq n+2}\). Examples are given to show that the upper bound is sharp for n ≥ 4, and the lower bound is sharp for n ≥ 8. Sharp lower bounds are also given for n = 4, 5, 6, 7, respectively.
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Chen, L., Li, X. & Lian, H. Nordhaus–Gaddum-Type Theorem for Rainbow Connection Number of Graphs. Graphs and Combinatorics 29, 1235–1247 (2013). https://doi.org/10.1007/s00373-012-1183-x
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DOI: https://doi.org/10.1007/s00373-012-1183-x