Cycles and paths in edge‐colored graphs with given degrees

Sufficient degree conditions for the existence of properly edge‐colored cycles and paths in edge‐colored graphs, multigraphs and random graphs are investigated. In particular, we prove that an edge‐colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to ⌈( n +1)/2⌉ has properly edge‐colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge‐colored paths and hamiltonian paths between given vertices are considered as well.