Rotation map

In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex $v$ and an edge label $i$, the rotation map returns the $i$'th neighbor of $v$ and the edge label that would lead back to $v$.

Definition

For a D-regular graph G, the rotation map $\mathrm{Rot}_G : [N] \times [D] \rightarrow [N] \times [D]$ is defined as follows: $\mathrm{Rot}_G (v,i)=(w,j)$ if the ith edge leaving v leads to w, and the jth edge leaving w leads to v.

Basic properties

From the definition we see that $\mathrm{Rot}_G$ is a permutation, and moreover $\mathrm{Rot}_G \circ \mathrm{Rot}_G$ is the identity map ($\mathrm{Rot}_G$ is an involution).

Special cases and properties

• A rotation map is consistently labeled if all of the edges leaving each vertex are labeled in such a way that at each vertex, the labels of the incoming edges are all distinct. Every regular graph has some consistent labeling.
• A rotation map is $\pi$-consistent if $\forall v \ \mathrm{Rot}_G(v,i)=(v[i],\pi (i))$. From the definition, a $\pi$-consistent rotation map is consistently labeled.

See also

• Zig-zag product
• Rotation system

References

• Reingold, O.; Vadhan, S.; Widgerson, A. (2000), "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors", 41st Annual Symposium on Foundations of Computer Science: 3–13, doi:10.1109/SFCS.2000.892006

• Reingold, O (2008), "Undirected connectivity in log-space", Journal of the ACM 55 (4): Article 17, 24 pages, doi:10.1145/1391289.1391291

• Reingold, O.; Trevisan, L.; Vadhan, S. (2006), "Pseudorandom walks on regular digraphs and the RL vs. L problem", Proceedings of the thirty-eighth annual ACM symposium on Theory of computing: 457, doi:10.1145/1132516.1132583