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PreprintCC BY 4.0arXiv

On a conjecture concerning antipodal labelings for cycles

  • John M. Campbell
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Abstract

Let G denote a finite, connected, simple graph, and let D denote its diameter, and let dG(u, v) denote the distance between vertices u and v in V(G). An antipodal labeling of G is a mapping f: V(G) → ℕ0 such that, for each pair $(u, v)$ consisting of distinct vertices in V(G), the relation $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ holds. The span of f is then defined so that sp(f) = max{ f(u) − f(v) : u, v ∈ V(G) }. The antipodal number of G, denoted with an(G), may then be defined as the minimum possible span among all antipodal labelings of G. Juan and Liu [Ars Combin., 2012] proved the upper bound an(C4k) ≤ 2k2 − 1, and conjectured that an(C4k) = 2k2 − 1 for each positive integer k, and Juan and Liu verified this conjecture for k ≤ 5. We succeed in proving Juan and Liu's conjecture, which seems to have remained open. The openness of Juan and Liu's conjecture has been noted, over the years, by many authors, including Rao et al. [Contrib. Discrete Math., 2015] and Saha et al. [Theory Comput. Syst., 2022].

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Cite

BibTeXRIS
BibTeX
@misc{campbell2026conjecture,
  title = {{On a conjecture concerning antipodal labelings for cycles}},
  author = {John M. Campbell},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29477},
  eprint = {2609.29477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29477},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On a conjecture concerning antipodal labelings for cycles
AU  - John M. Campbell
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.29477
UR  - https://arxiv.org/abs/2609.29477
AB  - Let G denote a finite, connected, simple graph, and let D denote its diameter, and let d_G(u, v) denote the distance between vertices u and v in V(G). An antipodal labeling of G is a mapping f: V(G) → ℕ_0 such that, for each pair $(u, v)$ consisting of distinct vertices in V(G), the relation $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ holds. The span of f is then defined so that sp(f) = max{ f(u) − f(v) : u, v ∈ V(G) }. The antipodal number of G, denoted with an(G), may then be defined as the minimum possible span among all antipodal labelings of G. Juan and Liu [Ars Combin., 2012] proved the upper bound an(C_4k) ≤ 2k^2 − 1, and conjectured that an(C_4k) = 2k^2 − 1 for each positive integer k, and Juan and Liu verified this conjecture for k ≤ 5. We succeed in proving Juan and Liu's conjecture, which seems to have remained open. The openness of Juan and Liu's conjecture has been noted, over the years, by many authors, including Rao et al. [Contrib. Discrete Math., 2015] and Saha et al. [Theory Comput. Syst., 2022].
ER  -