Fangfang Wu, Hajo Broersma, Shenggui Zhang, Binlong Li
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Properly colored and rainbow
C
4
${C}_{4}$
's in edge-colored graphs
We 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.
期刊介绍：
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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