Transversal factors and spanning trees

Given a collection of graphs $mathbf{G}=(G_1, ldots, G_m)$ with the same vertex set, an $m$-edge graph $Hsubset cup_{iin [m]}G_i$ is a transversal if there is a bijection $phi:E(H)to [m]$ such that $ein E(G_{phi(e)})$ for each $ein E(H)$. We give asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/log n)$.