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Properly coloured copies and rainbow copies of large graphs with small maximum degree

Let G be a graph on n vertices with maximum degree Δ. We use the Lovász local lemma to show the following two results about colourings χ of the edges of the complete graph Kn. If for each vertex v of Kn the colouring χ assigns each colour to at most (n ‐ 2)/(22.4Δ2) edges emanating from v, then there is a copy of G in Kn which is properly edge‐coloured by χ. This improves on a result of Alon, Jiang, Miller, and Pritikin [Random Struct. Algorithms 23(4), 409–433, 2003]. On the other hand, if χ assigns each colour to at most n/(51Δ2) edges of Kn, then there is a copy of G in Kn such that each edge of G receives a different colour from χ. This proves a conjecture of Frieze and Krivelevich [Electron. J. Comb. 15(1), R59, 2008].