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arXiv · 1812.10465

The number of Gallai k-colorings of complete graphs

Josefran de Oliveira Bastos·Fabricio S. Benevides·Jie Han
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Abstract

An edge coloring of the $n$-vertex complete graph, $K_n$, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for $n$ large and every $k$ with $k\le 2^{n/4300}$, the number of Gallai colorings of $K_n$ that use at most $k$ given colors is $(\binom{k}{2}+o_n(1))\,2^{\binom{n}{2}}$. Our result is asymptotically best possible and implies that, for those $k$, almost all Gallai $k$-colorings use only two colors. However, this is not true for $k \ge \Omega (2^{2n})$.

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Josefran de Oliveira Bastos, Fabricio S. Benevides, Jie Han. 2018-12-26. The number of Gallai k-colorings of complete graphs. https://arxiv.org/abs/1812.10465

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