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In [On the circumference of a graph and its complement, Discrete Math. 309 (2009), 5891–5893], Faudree et al. conjectured that when r≥3, every r-edge-colored complete graph Kn contains a monochromatic cycle of length at least n/(r−1). We disprove this conjecture for small n and give a short proof of the following weaker but more generalized form: for r≥1, every r-edge-colored complete graph Kn contains...
For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of Kn, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K3 and H is a cycle of a fixed length.
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