An edge-coloring of $G$ is $r$-bounded if every color appears in at most $r$ edges of $G$. A subgraph $H$ of an edge-colored graph $G$ is rainbow if all the edges of $H$ have a distinct color.
In this work we study the existence of rainbow perfect matching and Hamiltonian cycles under minimum degree conditions on an edge-colored graph where every color appears a bounded number of times. We derive asymptotically tight bounds on the minimum degree for the existence of such rainbow spanning structures.
The main ingredient in our proofs is to directly study the probability space given by the set of perfect matchings (or Hamiltonian cycles) of the graph $G$. This allows us to be more precise in the minimum degree condition needed to find the existence of such rainbow subgraph.
This is joint work with Guillem Perarnau and Oriol Serra.