Nim heaps in the Game of Cycles: the parity conjecture fails on trees
Abstract
The Game of Cycles, invented by Francis Su, is a two-player game in which the players take turns drawing arrows on the edges of a plane graph, never creating a sink or a source. On a tree only the sink-source rule matters, and the player who makes the last move wins. Several authors conjectured that on a tree the second player wins exactly when the number of markable edges is even. We show that this parity conjecture is false, and we explain why. The explanation is a hidden game of Nim: a path segment that is blocked by an arrow at one end and free at the other behaves exactly like a nim heap whose size is its length. Combined with a calculus for cutting and gluing trees at vertices next to leaves, this shows that the conjecture fails as soon as a single tree has Grundy value at least 2. The smallest counterexamples have 15 vertices. Trees with Grundy values 2 and 4 first appear at 17 vertices, and every value up to 63 occurs, which answers a question of Fokkink and Zandee. We also refute a related conjecture of Zandee. We then look closely at spiders. The spiders S(2,3,z) violate the conjecture for every z >= 14. The Grundy values of S(1,2j,z) are given by an exact formula governed by binary carries. For 99 pairs (x,y), the value of S(x,y,z) is eventually the parity of z. On the positive side, the conjecture holds for all trees in which every thread has at most two edges, and this bound is sharp. Along the way we prove a twin-branch reduction, give an algorithm that is polynomial for spiders with a fixed number of legs, and collect the questions that we could not answer. The boldest asks whether every non-negative integer is the Grundy value of some tree.
Related
- The Ramsey threshold for trees versus odd cycles
- Balanced even cycles in signed graphs:Turán bounds, double covers, and parity obstructions
- Digraph Laplacian spectra: The 2005 polygon conjecture fails for all orders above four
- Counting Weight-k Italian Dominating Sets on Trees and Cycles
- An Improved Bound for Smith's Longest Cycles Conjecture via a Forbidden Subdivision
BibTeX
@misc{xu2026heaps,
title = {{Nim heaps in the Game of Cycles: the parity conjecture fails on trees}},
author = {Muchen Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15145},
eprint = {2610.15145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15145},
}
RIS
TY - UNPB
TI - Nim heaps in the Game of Cycles: the parity conjecture fails on trees
AU - Muchen Xu
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15145
UR - https://arxiv.org/abs/2610.15145
AB - The Game of Cycles, invented by Francis Su, is a two-player game in which the players take turns drawing arrows on the edges of a plane graph, never creating a sink or a source. On a tree only the sink-source rule matters, and the player who makes the last move wins. Several authors conjectured that on a tree the second player wins exactly when the number of markable edges is even. We show that this parity conjecture is false, and we explain why. The explanation is a hidden game of Nim: a path segment that is blocked by an arrow at one end and free at the other behaves exactly like a nim heap whose size is its length. Combined with a calculus for cutting and gluing trees at vertices next to leaves, this shows that the conjecture fails as soon as a single tree has Grundy value at least 2. The smallest counterexamples have 15 vertices. Trees with Grundy values 2 and 4 first appear at 17 vertices, and every value up to 63 occurs, which answers a question of Fokkink and Zandee. We also refute a related conjecture of Zandee. We then look closely at spiders. The spiders S(2,3,z) violate the conjecture for every z >= 14. The Grundy values of S(1,2j,z) are given by an exact formula governed by binary carries. For 99 pairs (x,y), the value of S(x,y,z) is eventually the parity of z. On the positive side, the conjecture holds for all trees in which every thread has at most two edges, and this bound is sharp. Along the way we prove a twin-branch reduction, give an algorithm that is polynomial for spiders with a fixed number of legs, and collect the questions that we could not answer. The boldest asks whether every non-negative integer is the Grundy value of some tree.
ER -