Sufficient conditions for properly colored C₃’s and C₄’s in edge-colored complete graphs

For an edge-colored graph, its minimum color degree is the minimum number of distinct colors appearing on the edges incident with a vertex, and its maximum monochromatic degree is the maximum number of edges with the same color incident with a vertex. A cycle in an edge-colored graph is called properly colored if any two consecutive edges of the cycle have distinct colors. We investigate sufficient conditions in terms of the minimum color degree and maximum monochromatic degree for the existence of short properly colored cycles in edge-colored complete graphs. In particular, we obtain sharp results for the existence of properly colored C₄’s, and we characterize the extremal graphs for several known results on the existence of properly colored triangles. Moreover, we obtain sharp sufficient conditions guaranteeing that every vertex is contained in a properly colored triangle or C₄, respectively.