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Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes — arXiv:2609.02585

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.02585
License (as archived): CC0-1.0
Archived: 2026-10-17T05:38:46.581Z
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How to cite

Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes — arXiv:2609.02585 — arXiv (en), https://arxiv.org/abs/2609.02585 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bc857b60af35 (Metraipolis archive dc_bc857b60af35, sha256 d94c5242b6e3d8826d7cf0b4914247ddf74849455ad51820222d7c3dc046d568)

BibTeX

@misc{dcbc857b60af35,
  title = {Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes — arXiv:2609.02585},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.02585},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bc857b60af35; sha256 d94c5242b6e3d8826d7cf0b4914247ddf74849455ad51820222d7c3dc046d568},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.02585. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Although almost all graphs are asymmetric -- having no nontrivial global automorphisms -- they may still possess local symmetries in the form of isomorphisms between induced subgraphs, i.e., partial automorphisms. We study such local symmetries via asymmetric depth, defined in terms of the maximum rank of a nontrivial partial automorphism. We prove a tight upper bound on asymmetric depth in the class of planar graphs and identify the extremal graphs: duals of IPR fullerenes attain the maximum already on $47$ vertices. Our main structural result concerns the IPR fullerenes that are neither maximally asymmetric nor symmetric. In such a cage no purely local action realises a low asymmetric depth, and we show that the map which does realise it cannot be confined to a small part of the cage either: neither to a single face, nor behind an interface of at most $5-k$ edges, $k \le 3$ being the deficiency. A cage of asymmetric depth $2$ or $3$ is therefore not asymmetric in one place; it carries a broken symmetry invisible to its automorphism group. Such cages are rare -- under $2\%$ of the asymmetric IPR fullerenes at $n = 118$. In all $727$ of them the largest partial automorphism is a near-mirror reflection, which we state as an explicit conjecture. We also extend the asymmetric depth bound to graphs of higher genus.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-03 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Discrete Mathematics (cs.DM); Group Theory (math.GR).
Pagina del lavoro: https://arxiv.org/abs/2609.02585 — 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 2026-10-17T04:59:51.785Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-03?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).