$n$-叉树的定位-色数
The Locating-Chromatic number of an $n$-ary Trees
AI总结：
本文研究了$k$层$n$-叉树定位-色数的渐近值，发现当固定层数$k$趋于无穷大时色数趋于$n+k-1$，而固定叉数$n$趋于无穷大时色数为$o(k)$，揭示了两种极限下的截然不同表现。
摘要：
图$G$的定位-色数是最小整数$n$，使得$G$有一个真$n$-染色$c$，且所有顶点到$c$生成的颜色的距离向量各不相同。我们研究了$k$层$n$-叉树定位-色数的渐近值。当$k$趋于无穷大和$n$趋于无穷大时，该树的定位-色数表现截然不同。若固定$k\geq2$，几乎所有$n$-叉树$T(n,k)$满足$χ_L(T(n,k))=n+k-1$；故$\lim\limits_{n\to \infty} χ_L(T(n,k))-n=k-1$。但若固定$n\geq 2$，则$χ_L(T(n,k))=o(k)$。
英文摘要：
The locating-chromatic number of a graph $G$ is the smallest integer $n$, such that $G$ has a proper $n$-coloring $c$ and all vertices have different vectors of distances to the colors generated by $c$. We study the asymptotic value of the locating-chromatic number of a $k$-level $n$-ary tree. The locating-chromatic number of this tree acts very differently when $k$ goes to infinity and when $n$ goes to infinity. If we fix $k\geq2$, almost all $n$-ary Tree $T(n,k)$ satisfy $χ_L(T(n,k))=n+k-1$; so $\lim\limits_{n\to \infty} χ_L(T(n,k))-n=k-1$. But if we fix $n\geq 2$, then $χ_L(T(n,k))=o(k)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
