Čada, Roman; Kaneko, Atsushi; Ryjáček, Zdeněk; Yoshimoto, Kiyoshi Rainbow cycles in edge-colored graphs. (English) Zbl 1329.05103 Discrete Math. 339, No. 4, 1387-1392 (2016). 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. Cited in 10 Documents MSC: 05C15 Coloring of graphs and hypergraphs Keywords:edge coloring; rainbow coloring; rainbow cycle; color degree condition PDFBibTeX XMLCite \textit{R. Čada} et al., Discrete Math. 339, No. 4, 1387--1392 (2016; Zbl 1329.05103) Full Text: DOI References: [1] Bondy, J. A.; Murty, U. S.R., Graph Theory (2008), Springer · Zbl 1134.05001 [2] Broersma, H. J.; Li, X.; Woeginger, G.; Zhang, S., Paths and cycles in colored graphs, Australas. J. Combin., 31, 299-311 (2005) · Zbl 1061.05030 [3] Caccetta, L.; Häggkvist, R., On minimal digraphs with given girth, Congr. Numer., 21, 181-187 (1978) · Zbl 0406.05033 [4] Erdös, P.; Gallai, T., On maximal paths and circuits in graphs, Acta Math. Sci. Hungar., 10, 337-356 (1959) · Zbl 0090.39401 [5] Kano, M.; Li, X., Monochromatic and heterochromatic subgraphs in edge-colored graphs — a survey, Graphs Combin., 24, 237-263 (2008) · Zbl 1190.05045 [6] Li, H., Rainbow \(C_3\)’s and \(C_4\)’s in edge-colored graphs, Discrete Math., 313, 1893-1896 (2013) · Zbl 1277.05065 [7] Li, H.; Wang, G., Color degree and heterochromatic cycles in edge-colored graphs, European J. Combin., 33, 1958-1964 (2012) · Zbl 1248.05071 [8] Li, B.; Xu, C.; Zhang, S., Rainbow triangles in edge-colored graphs, European J. Combin., 36, 453-459 (2014) · Zbl 1284.05103 [9] Wang, G.; Li, H.; Zhu, Y.; Liu, G., A note on heterochromatic \(C_4\) in edge-colored triangle-free graphs, Graphs Combin., 28, 901-905 (2012) · Zbl 1256.05080 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.