Stability of transversal Hamilton cycles and paths

Given graphs G₁,…,Gₛ all on a common vertex set and a graph H with $e(H) = s$, a copy of H is transversal or rainbow if it contains one edge from each Gᵢ. We establish a stability result for transversal Hamilton cycles: the minimum degree required to guarantee a transversal Hamilton cycle can be lowered as long as the graph collection G₁,…,Gₙ is far in edit distance from several extremal cases. We obtain an analogous result for Hamilton paths. The proof is a combination of our newly developed regularity-blow-up method for transversals, along with the absorption method.