Gallai-Colorings of Triples and 2-Factors of B[subscript 3]
Author(s)Chua, Lynn; Gyarfas, Andras; Hossain, Chetak
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A coloring of the edges of the r-uniform complete hypergraph is a G[subscript r]-coloring if there is no rainbow simplex; that is, every set of r + l vertices contains two edges of the same color. The notion extends G[subscript 2]-colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of G[subscript 2]-colorings is that at least one color class has a spanning tree. J. Lehel and the senior author observed that this property does not hold for G[subscript r]-colorings and proposed to study f[subscript r](n), the size of the largest monochromatic component which can be found in every G[subscript r]-coloring of K[r over n], the complete r-uniform hypergraph. The previous remark says that f[subscript 2](n) = n, and in this note, we address the case r = 3. We prove that [(n + 3)/2] ≤ f[subscript 3](n) ≤ [4n/5], and this determines f[subscript 3](n) for n < 7. We also prove that f[subscript 3](7) = 6 by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
International Journal of Combinatorics
Hindawi Publishing Corporation
Chua, Lynn, Andras Gyarfas, and Chetak Hossain. “Gallai-Colorings of Triples and 2-Factors of B[subscript 3].” International Journal of Combinatorics 2013 (2013): 1–6.
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