Exact-palette rainbow embeddings in uniformly coloured pseudorandom graphs

Abstract

We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,β)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficient palette surplus of order $(\log n)/p$, assuming $pn=ω(\log n)$ and $β\le cpn$ for a sufficiently small constant $c$. For each prescribed spanning tree of fixed maximum degree $Δ\ge2$, a surplus of order $L_{n,Δ}(\log n)/p$ suffices under $β\le cpn/L_{n,Δ}$, where $L_{n,Δ}=Δ^{5\sqrt{\log n}}$. The palette surplus is sublinear under these hypotheses. These results use a McDiarmid-type coupling and retain the discrepancy scales of the relevant deterministic embedding theorems. We also prove exact-palette results, using precisely as many colours as the number of edges of the target configuration. After independent edge percolation at rate $ρ$, it is shown that a rainbow perfect matching or Hamilton cycle exists asymptotically almost surely when $ρpn \ge C(\log n)^2$ and $β\leγpn$ for appropriate constants $C$ and $γ$. We obtain corresponding results for each prescribed bounded-degree spanning tree and for clique factors under appropriate stronger hypotheses. The exact-palette proofs construct spread measures from uncoloured containment estimates and apply the rainbow threshold theorem of Han and Yuan.