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Rainbow cycles in edge-colored graphs. (English) Zbl 1329.05103

Summary: Let \(G\) be a graph of order \(n\) with an edge coloring \(c\), and let \(\delta^c(G)\) denote the minimum color degree of \(G\), i.e., the largest integer such that each vertex of \(G\) is incident with at least \(\delta^c(G)\) edges having pairwise distinct colors. A subgraph \(F \subset G\) is rainbow if all edges of \(F\) have pairwise distinct colors. In this paper, we prove that (i) if \(G\) is triangle-free and \(\delta^c(G) > \frac{n}{3} + 1\), then \(G\) contains a rainbow \(C_4\), and (ii) if \(\delta^c(G) > \frac{n}{2} + 2\), then \(G\) contains a rainbow cycle of length at least 4.

MSC:

05C15 Coloring of graphs and hypergraphs
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