Rainbow structures in a collection of graphs with degree conditions

Abstract

AbstractLet be a collection of not necessarily distinct ‐vertex graphs with the same vertex set . We use to denote an edge‐colored multigraph of with and a multiset consisting of , and the edge of is colored by if . A graph is rainbow in if any two edges of belong to different graphs of . We say that is rainbow vertex‐pancyclic if each vertex of is contained in a rainbow cycle of with length for every integer , and that is rainbow panconnected if for any pair of vertices and of there exists a rainbow path of with length joining and for every integer . In this paper, we study the existences of rainbow spanning trees and rainbow Hamiltonian paths in under the Ore‐type conditions. Moreover, we study the rainbow vertex‐pancyclicity and rainbow panconnectedness, as well as the existence of rainbow cliques in under the Dirac‐type conditions. We also give some examples to show the sharpness of our results.