Properly colored and rainbow C₄' S in edge-colored graphs

We present new sharp sufficient conditions for the existence of properly colored and rainbow (Formula presented.) 's in edge-colored graphs. Our first results deal with sharp color neighborhood conditions for the existence of properly colored (Formula presented.) 's in edge-colored complete graphs and complete bipartite graphs, respectively. Next, we characterize the extremal graphs for an anti-Ramsey number result due to Alon on the existence of rainbow (Formula presented.) 's in edge-colored complete graphs. We also generalize Alon's result from complete to general edge-colored graphs. Finally, we derive a structural property regarding the extremal graphs for a bipartite counterpart of Alon's result due to Axenovich, Jiang, and Kündgen on the existence of rainbow (Formula presented.) 's in edge-colored complete bipartite graphs. We also generalize their result from complete to general bipartite edge-colored graphs.