Ore's Theorem for rainbow Hamiltonian-connected graphs

Let $G = (G_1, G_2, \ldots, G_m)$ be a collection of $m$ graphs on a common vertex set $V$. For a graph $H$ with vertices in $V$, we say that $G$ contains a rainbow $H$ if there is an injection $c: E(H) \to [m]$ such that for every edge $e \in E(H)$, we have $e \in E(G_{c(e)})$. In this paper, we show that if $G = (G_1, \ldots, G_n)$ is a collection of graphs on $n$ vertices such that for every $i \in [n]$, $d_{G_i}(u) + d_{G_i}(v) \geq n$ whenever $uv \notin E(G_i)$, then either $G$ contains rainbow Hamiltonian paths between every pair of vertices, or $G$ contains a rainbow Hamiltonian cycle. Moreover, we prove a stronger version in which we may also embed prescribed rainbow linear forests into the Hamiltonian paths.