A rainbow blow-up lemma for almost optimally bounded edge-colourings

Abstract

A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi that applies for almost optimally bounded edge-colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that can be seen to be almost best possible. This has many interesting applications beyond rainbow colourings, for example to graph decompositions. There are several well-known conjectures in graph theory concerning tree decompositions, such as Kotzig's conjecture and Ringel's conjecture. We adapt these conjectures to general bounded-degree subgraphs, and provide asymptotic solutions using our result on rainbow embeddings.