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 Komlós, Sárközy, and Szemerédi that applies to almost optimally bounded 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 is asymptotically best possible. This has many applications beyond rainbow colourings: for example, to graph decompositions, orthogonal double covers, and graph labellings.

Pseudorandom hypergraph matchings

“…In Section 4.1 we deduce the existence of approximate Steiner systems that behave 'randomly' , for example with respect to subgraph statistics. Then we briefly explain how we apply Theorem 1.3 in two forthcoming papers [6,7] on rainbow embeddings and approximate decompositions.…”

“…In [6] we consider subgraph embeddings in edge-coloured graphs with the additional requirement that the embedded subgraph is 'rainbow' , meaning that any two edges in the subgraph have distinct colours. Such rainbow embeddings have applications to various other problems.…”

Pseudorandom hypergraph matchings

“…with respect to subgraph statistics. Then, we briefly explain how we apply it in two forthcoming papers [6,7] on rainbow embeddings and approximate decompositions. 4.1.…”

“…Rainbow problems. In [6], we consider subgraph embeddings in edge-coloured graphs with the additional requirement that the embedded subgraph is 'rainbow', meaning that any two edges in the subgraph have distinct colours. Such rainbow embeddings have applications to various other problems.…”

The Rainbow Connection Number of Triangular Snake Graphs

“…We address the problem of determining the rainbow connection number of a graph, introduced more than 10 years ago [Chartrand et al 2008]. Rainbow colorings appear in many different contexts of Combinatorics [Ehard et al 2019, Montgomery et al 2019]. Let k be a non-negative integer and c : E(G) → {0, 1, .…”