For an arbitrary finite family of graphs, the distance labeling problem asks to assign labels to all nodes of every graph in the family in a way that allows one to recover the distance between any two nodes of any graph from their labels. The main goal is to minimize the number of unique labels used. We study this problem for the families $\mathcal{C}_n$ consisting of cycles of all lengths between 3 and $n$. We observe that the exact solution for directed cycles is straightforward and focus on the undirected case. We design a labeling scheme requiring $\frac{n\sqrt{n}}{\sqrt{6}}+O(n)$ labels, which is almost twice less than is required by the earlier known scheme. Using the computer search, we find an optimal labeling for each $n\le 17$, showing that our scheme gives the results that are very close to the optimum.

翻译：对于任意有限图族，距离标号问题要求为该族中每个图的所有节点分配标号，使得能从任意两个节点的标号恢复出它们在图中距离。主要目标是尽可能减少使用唯一标号的数量。本文研究由长度在3到n之间的所有环构成的族$\mathcal{C}_n$的距离标号问题。我们注意到有向环的精确解是直接的，并重点关注无向情形。我们设计了一种标号方案，需要$\frac{n\sqrt{n}}{\sqrt{6}}+O(n)$个标号，比此前已知方案所需的标号数量减少近一半。通过计算机搜索，我们给出了每个$n\le 17$的最优标号，表明我们的方案所得结果非常接近最优值。