Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems

Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\in V$, set \[ δ(v)=\min_{i\in[n]}d_{G_i}(v), \qquad E_{\cap}=\bigcap_{i=1}^n E(G_i), \] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$--$v$ path. We study Hamiltonicity conditions imposed only on pai