Abstract
A new proof of the following well-known theorem due to Fan is given: Let G be a 2-connected graph of order n, and 3 ≤ c ≤ n. Ifmax{d(u), d(v)} ≥ c/2 for any two vertices u, v at distance 2, then G contains a cycle of length at least c.
References
[1] BONDY, J. A.: Large cycles in graphs, Discrete Math. 1 (1971), 121-132.10.1016/0012-365X(71)90019-7Search in Google Scholar
[2] FAN, G. H.: New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984), 221-227.10.1016/0095-8956(84)90054-6Search in Google Scholar
[3] GAGO, S.-HURAJOV´A, J.-MADARAS, T.: Notes on the betweenness centrality of a graph, Math. Slovaca 62 (2012), 1-12.10.2478/s12175-011-0065-7Search in Google Scholar
[4] TIAN, F.: A short proof of a theorem about the circumference of a graph, J. Combin. Theory Ser. B 45 (1988), 373-375.10.1016/0095-8956(88)90080-9Search in Google Scholar
[5] TIAN, F.: A short proof of Fan’s theorem, Discrete Math. 286 (2004), 285-286.10.1016/j.disc.2004.05.012Search in Google Scholar
[6] VOLKMANN, L.: Bounds on the k-tuple domatic number of a graph, Math. Slovaca 61 (2011), 851-858.10.2478/s12175-011-0052-zSearch in Google Scholar
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