Distance Labeling for Families of Cycles

TL;DR

This paper develops a near-optimal distance labeling scheme for families of undirected cycles, significantly reducing label count compared to previous methods, with computational verification for small cycle lengths.

Contribution

It introduces a new labeling scheme for cycle families that nearly halves the label count of prior schemes and provides computationally verified optimal labels for small cycles.

Findings

The scheme requires approximately n√n/√6 labels, improving over previous methods.

Optimal labels for cycles up to length 17 are identified, closely matching the scheme's performance.

The exact solution for directed cycles is straightforward, focusing the contribution on undirected cases.

Abstract

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.