On the identification problems in products of cycles

Abstract

A set of subgraphs G1,…,Gk in a graph G is said to identify the vertices v (resp. edges e) if the sets {j:v∈Gj} (resp. {j:e∈Gj}) are all nonempty and different. In this paper we prove upper bounds for the smallest cardinalities of vertex and edge identifying collections of cycles and closed walks. In particular, we prove that the smallest cardinality of edge identifying collection of closed walks in the binary Hamming space is n+⌊log2n⌋. We also consider the identification of paths of length two.