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

We present new sharp sufficient conditions for the existence of properly colored and rainbow C4's in edge -colored graphs. Our first results deal with sharp color neighborhood conditions for the existence of properly colored C-4'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 C-4'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 Kundgen on the existence of rainbow C4's in edge -colored complete bipartite graphs. We also generalize their result from complete to general bipartite edge -colored graphs.