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Disjoint union activation of difference distance magic labelings

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Daniel Adam Pyle ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

A graph is difference distance magic orientable (DDMO) if its vertices can be labeled bijectively by consecutive positive integers and its edges oriented so that the incoming and outgoing neighbor-label sums agree at every vertex. We give an affirmative answer to the disjoint-union question posed by Aceska et al. in January 2026: two copies of the non-DDMO graph K₃,₃ have a DDMO disjoint union. More precisely, for k ≥ 1 and r ≥ 0, the graph consisting of k disjoint copies of K₃,₃ and r isolated vertices is difference distance magic orientable if and only if k is even or r ≥ 1. The proof reduces fixed-label bipartite orientations to signed zero sums and applies classical Skolem and hooked Skolem pairings. A compatible fixed labeling of K₃,₃ has exactly two orientations, related by reversal. We prove that twelve vertices is the minimum order of a DDMO graph whose connected components are all non-DDMO in isolation. A separate exact computer-assisted classification shows that two disjoint copies of K₃,₃ form the unique graph attaining that minimum.

Provenance statement

The literature research, proof development, computational verification, and manuscript drafting were performed by an OpenAI-powered AI assistant called Lyra. Daniel A. Pyle initiated and directed the project and is submitting the work as an independent researcher. Verification included separate AI-assisted reviews and two exact computational classifications. These checks do not constitute human peer review or proof-assistant formalization. Classical constructions and prior results are credited to their sources. A broader extension is explicitly conditional on a stated external theorem whose proof was not fully verified. Bibliographic novelty remains provisional. The project repository for the manuscript and verification materials is https://github.com/dapyle11/Disjoint-union-activation-of-difference-distance-magic-labelings.

Tools used

OpenAI
OpenAI-powered assistant (Lyra) Python and SymPy.

References

  1. R. Aceska, N. Arcila-Maya, J. Carlson, A. Marr, M. Parnes, K. Ryan, H. Schuerger, and J. F. Vasquez. New results on difference distance magic labelings. arXiv:2601.20492v1, 2026. Definition 2.1; Theorem 5.1; union question in Section 6, p. 19. Full text: https://arxiv.org/pdf/2601.20492v1.arXiv
  2. K. Altman, L. Calzado, B. Freyberg, P. Kovář, E. Lewis, A. Marr, L. Ross, and R. Via. Difference distance magic oriented graphs. Research in the Mathematical Sciences 11 , article 67, 2024. DOI: 10.1007/s40687-024-00482-7 (https://doi.org/10.1007/s40687-024-00482-7). Complete bipartite characterization: Theorem 5, p. 4. Full text: https://d-nb.info/1355962609/34.DOI
  3. Th. Skolem. On certain distributions of integers in pairs with given differences. Mathematica Scandinavica 5 (1957), 57--68. DOI: 10.7146/math.scand.a-10490 (https://doi.org/10.7146/math.scand.a-10490). Theorems 1 and 2, p. 57; order-4m4m construction, p. 58. Full text: https://journals.msp.org/mscand/article/download/2833/2832/2864.DOI
  4. E. S. O'Keefe. Verification of a conjecture of Th. Skolem. Mathematica Scandinavica 9 (1961), 80--82. DOI: 10.7146/math.scand.a-10624 (https://doi.org/10.7146/math.scand.a-10624). Theorem and order-4m+24m+2 construction, pp. 80--81. Full text: https://journals.msp.org/mscand/article/download/2721/2720/2752.DOI
  5. M. Licona and J. Tey. Gracefulness of two nested cycles: a first approach. arXiv:2411.12998v1, 2024. Definition 4.1 and Lemma 3.17, p. 13; Theorem 4.17, p. 23. Full text: https://arxiv.org/pdf/2411.12998v1.arXiv

Version history

  1. v1 · 9 pagesSubmitted by Daniel Adam PyleInitial depositCurrentOct 20, 2026