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