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PreprintarXiv licencearXiv

A Counterexample to Wormald's Conjecture

  • James Alexander Schreib
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Abstract

We give a counterexample to Wormald's conjecture. A 16-vertex cubic graph, the disjoint union of K3,3 with a 10-vertex bridged graph built from two subdivided copies of K4, has no partition of its edges into two isomorphic spanning linear forests. The obstruction is parity: any such partition would force every monochromatic component to have even order while requiring a five-vertex side of a bridge to be partitioned into such components. Adjoining copies of K4 gives counterexamples in every order 16+4t. The counterexample is disconnected and leaves the connected case unresolved. The full family is verified in Lean without custom axioms.

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Cite

BibTeXRIS
BibTeX
@misc{schreib2026counterexample,
  title = {{A Counterexample to Wormald's Conjecture}},
  author = {James Alexander Schreib},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00086},
  eprint = {2610.00086},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00086},
}
RIS
TY  - UNPB
TI  - A Counterexample to Wormald's Conjecture
AU  - James Alexander Schreib
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00086
UR  - https://arxiv.org/abs/2610.00086
AB  - We give a counterexample to Wormald's conjecture. A 16-vertex cubic graph, the disjoint union of K_3,3 with a 10-vertex bridged graph built from two subdivided copies of K_4, has no partition of its edges into two isomorphic spanning linear forests. The obstruction is parity: any such partition would force every monochromatic component to have even order while requiring a five-vertex side of a bridge to be partitioned into such components. Adjoining copies of K_4 gives counterexamples in every order 16+4t. The counterexample is disconnected and leaves the connected case unresolved. The full family is verified in Lean without custom axioms.
ER  -