Gallai-Colorings of Triples and 2-Factors of B[subscript 3] Citation
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The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. A coloring of the edges of the í µí±-uniform complete hypergraph is a í µí°º í µí±-coloring if there is no rainbow simplex; that is, every set of í µí± + 1 vertices contains two edges of the same color. The notion extends í µí°º 2-colorings which are often called Gallai-colorings and originates from a seminal paper of Gallai. One well-known property of í µí°º 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 í µí°º í µí±-colorings and proposed to study í µí± í µí± (í µí±), the size of the largest monochromatic component which can be found in every í µí°º í µí±-coloring of í µí°¾ í µí± í µí± , the complete í µí±-uniform hypergraph. The previous remark says that í µí± 2 (í µí±) = í µí±, and in this note, we address the case í µí± = 3. We prove that ⌈(í µí± + 3)/2⌉ ≤ í µí± 3 (í µí±) ≤ ⌈4í µí±/5⌉, and this determines í µí± 3 (í µí±) for í µí± < 7. We also prove that í µí± 3 (7) = 6 by excluding certain 2-factors from the middle layer of the Boolean lattice on seven elements.
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Gallai colorings of non-complete graphs
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