Rainbow structures in a collection of graphs with degree conditions

Let G={G1, horizontal ellipsis ,Gs} ${\bf{G}}=\{{G}_{1},\ldots ,{G}_{s}\}$ be a collection of not necessarily distinct n $n$-vertex graphs with the same vertex set V $V$. We use G similar to $\tilde{{\bf{G}}}$ to denote an edge-colored multigraph of G ${\bf{G}}$ with V(G similar to)=V $V(\tilde{{\bf{G}}})=V$ and E(G similar to) $E(\tilde{{\bf{G}}})$ a multiset consisting of E(G1), horizontal ellipsis ,E(Gs) $E({G}_{1}),\ldots ,E({G}_{s})$, and the edge e $e$ of G similar to $\tilde{{\bf{G}}}$ is colored by i $i$ if e is an element of E(Gi) $e\in E({G}_{i})$. A graph H $H$ is rainbow in G ${\bf{G}}$ if any two edges of H $H$ belong to different graphs of G ${\bf{G}}$. We say that G ${\bf{G}}$ is rainbow vertex-pancyclic if each vertex of V $V$ is contained in a rainbow cycle of G ${\bf{G}}$ with length l $\ell $ for every integer l is an element of[3,n] $\ell \in [3,n]$, and that G ${\bf{G}}$ is rainbow panconnected if for any pair of vertices u $u$ and v $v$ of V $V$ there exists a rainbow path of G ${\bf{G}}$ with length l $\ell $ joining u $u$ and v $v$ for every integer l is an element of[dG similar to(u,v),n-1] $\ell \in [{d}_{\tilde{{\bf{G}}}}(u,v),n-1]$. In this paper, we study the existences of rainbow spanning trees and rainbow Hamiltonian paths in G ${\bf{G}}$ under the Ore-type conditions. Moreover, we study the rainbow vertex-pancyclicity and rainbow panconnectedness, as well as the existence of rainbow cliques in G ${\bf{G}}$ under the Dirac-type conditions. We also give some examples to show the sharpness of our results.