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The edge multiset dimension of hypercubes — arXiv:2608.09983

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.09983
License (as archived): CC0-1.0
Archived: 2026-10-19T06:27:19.891Z
SHA-256 of the archived text: a86d8645e4574ba149ccda62e459e3b6e381784dfb3af413923b5959eaf440fa
Stable URL of this page: https://metraipolis.com/documento/dc_55fd66f855fd

How to cite

The edge multiset dimension of hypercubes — arXiv:2608.09983 — arXiv (en), https://arxiv.org/abs/2608.09983 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_55fd66f855fd (Metraipolis archive dc_55fd66f855fd, sha256 a86d8645e4574ba149ccda62e459e3b6e381784dfb3af413923b5959eaf440fa)

BibTeX

@misc{dc55fd66f855fd,
  title = {The edge multiset dimension of hypercubes — arXiv:2608.09983},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.09983},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_55fd66f855fd; sha256 a86d8645e4574ba149ccda62e459e3b6e381784dfb3af413923b5959eaf440fa},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.09983. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-12 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.09983 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T06:20:30.677Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-12?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).