Long rainbow paths and rainbow cycles in edge colored graphs - A survey

Graph coloring problem and problem on the existence of paths and cycles have always been popular topics in graph theory. The problem on the existence of rainbow paths and rainbow cycles in edge colored graphs, as an integration of them, was well studied for a long period. In this survey, we will review known results on this subject. Because of the relationship between cycles and paths, we will review results on the existence of rainbow cycles (including rainbow Hamilton cycles, long rainbow cycles and rainbow cycles with given length) first, and then long rainbow paths (including rainbow Hamilton paths and other long rainbow paths).