避免彩虹 $K_4$ 的完全图边着色计数
Counting edge-colorings of a complete graph avoiding a rainbow $K_4$
- Universidade Federal do Ceará (UFC)（塞阿拉联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究避免彩虹 $K_4$ 的完全图边着色计数，证明对 $r\ge25$ 大多数着色使用超过五种颜色，并给出指数增长率，结合容器法与图移除引理。
AI中文摘要：
对于自然数 $k, r, n$，令 $\rho_{r,k}(K_n)$ 为 $K_n$ 的不含彩虹 $K_k$ 副本（即所有边颜色均不同的 $K_k$ 副本）的 $r$-边着色数。当 $k=3$ 时，$\rho_{r,3}(K_n)$ 表示 Gallai 着色的数量。Balogh 和 Li 以及独立地 Bastos, Benevides 和 Han 证明了：当 $n$ 足够大时，大多数 Gallai 着色是 2-着色。一个自然的类比猜想是：当 $k=4$，$r\ge 5$ 且 $n$ 足够大时，大多数无彩虹-$K_4$ 的 $r$-边着色是 5-着色。我们证明这通常不成立，并确定了使该猜想不再成立的 $r$ 的精确阈值。对于猜想不成立的 $r$ 的范围，我们确定了每个固定 $r$ 下 $\rho_{r,4}(K_n)$ 的指数增长。更精确地，对于 $6\le r\le24$，我们证明 $\rho_{r,4}(K_n)=(\binom{r}{5}+o(1))5^{\binom{n}{2}}$；对于每个 $r\ge25$，使用至多五种颜色的比例趋于零，且 $\rho_{r,4}(K_n)=r^{(n^2/4)+o(n^2)}$。一个二分构造，其中两个部分内的所有边都赋予一种共同颜色，实现了后者的指数增长率。下界可以容易地推广到每个 $k$。这些结果与最近关于避免彩虹团或给定彩虹图案的着色计数的其他结果相关。我们的证明结合了超图容器、图移除引理、颜色调色板的结构估计以及对接近固定五色调色板的着色的精细计数。
英文摘要：
For $k, r, n$ natural numbers let $ρ_{r,k}(K_n)$ be the number of $r$-edge-colorings of $K_n$ that do not contain a rainbow copy of a $K_k$, that is, a copy of $K_k$ in which all edges receive different colors. When $k=3$, the quantity $ρ_{r,3}(K_n)$ represents the number of Gallai Colorings. It was proved by Balogh and Li and independently by Bastos, Benevides and Han, that most of the Gallai colorings are 2-colorings, for $n$ large. A natural analogue conjecture would be that when $k=4$, $r\ge 5$ and $n$ large, most rainbow-$K_4$-free $r$-edge-colorings are $5$-colorings. We show that this is not true in general and identify an exact threshold for $r$ where this ceases to be true. For the range where the conjecture is false, we determine the exponential growth of $ρ_{r,4}(K_n)$ for every fixed $r$. More precisely, for \(6\le r\le24\), we prove that \(ρ_{r,4}(K_n)=(\binom{r}{5}+o(1))5^{\binom{n}{2}}\); and for each \(r\ge25\), the proportion using at most five colors tends to zero, and \(ρ_{r,4}(K_n)=r^{(n^2/4)+o(n^2)}\). A bipartite construction, with all edges within the two parts assigned one common color, achieves the latter exponential growth rate. The lower bounds can be easily generalized for every $k$. Those results are related to other recent results about counting colorings that avoid rainbow cliques or given rainbow patterns in general. Our proof combines hypergraph containers with the graph removal lemma, structural estimates for color palettes and a refined count of colorings close to a fixed five-color palette.
