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A Winner Classification for Rooted Partizan Edge Geography on Heterogeneous Bouquets of Cycles
Abstract
We consider an undirected edge-traversal game on a finite bouquet of simple cycles. Each of two players controls a token, both tokens begin at the common root, and the players alternate traversing edges that have not been used by either token. A player with no legal move loses. We prove that the first player wins precisely when the number of cycles of odd edge-length is odd, without assuming equal cycle lengths. The proof uses a simultaneous induction on root and defense configurations, with an exact analysis of the forced race between two opened cycles. We also explain how the classification corresponds to a completed Lean 4 formalization of the underlying game on finite edge sets.
Related
BibTeX
@misc{tilepe2026winner,
title = {{A Winner Classification for Rooted Partizan Edge Geography on Heterogeneous Bouquets of Cycles}},
author = {Adam Tilèpe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40683},
eprint = {2610.40683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40683},
}
RIS
TY - UNPB
TI - A Winner Classification for Rooted Partizan Edge Geography on Heterogeneous Bouquets of Cycles
AU - Adam Tilèpe
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.40683
UR - https://arxiv.org/abs/2610.40683
AB - We consider an undirected edge-traversal game on a finite bouquet of simple cycles. Each of two players controls a token, both tokens begin at the common root, and the players alternate traversing edges that have not been used by either token. A player with no legal move loses. We prove that the first player wins precisely when the number of cycles of odd edge-length is odd, without assuming equal cycle lengths. The proof uses a simultaneous induction on root and defense configurations, with an exact analysis of the forced race between two opened cycles. We also explain how the classification corresponds to a completed Lean 4 formalization of the underlying game on finite edge sets.
ER -