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 inverstigated. 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 ⌈ 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.

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