The edge multiset dimension of hypercubes

Abstract

For a graph G and a nonempty set S of vertices, the edge multiset representation of an edge e is the multiset of distances from e to the elements of S, where d(uv,s)=min{d(u,s),d(v,s)}. The edge multiset dimension edim_m(G) is the minimum cardinality of a set whose edge representations are pairwise distinct, and is infinite if no such set exists. A recent survey asked whether edim_m(Q_d) is infinite for every d >= 3. We answer this question negatively and determine the finite-infinite transition completely: edim_m(Q_d) is infinite if and only if 2 <= d <= 5. An exhaustive computation proves edim_m(Q_5) = infinity, extending the known nonexistence results for Q_3 and Q_4. Explicit independently verifiable resolving sets are given for Q_6 through Q_10. For all d >= 11 we prove existence probabilistically: equality of two random edge histograms is a zero-divergence event on a graph of distance levels, and conditioning outside a spanning forest bounds its probability by a product of central-binomial atoms. Certified exact rational computations cover 11 <= d <= 50, and an elementary ten-edge forest estimate handles the tail d >= 51. We also prove the lower bound edim_m(Q_6) >= 6.