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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.20184 |
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| _version_ | 1866911725957152768 |
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| author | Hollom, Lawrence |
| author_facet | Hollom, Lawrence |
| contents | What is the maximum, over all 2-colourings of the edges of the $n$-dimensional hypercube $Q_n$, of the minimal number of times a path between a vertex $v$ and its antipode $\bar{v}$ changes colour? A conjecture of Norine, in a form due to Feder and Subi, states that this maximum should be 1. The previous best-known upper bound on the number of colour changes was $(\tfrac{5}{16} + o(1))n$ due to Kirchweger, Peitl, Subercaseaux, and Szeider. We improve this bound and answer a question of Leader and Long by finding a geodesic path with at most $(\tfracπ{2} + o(1))\sqrt{n}$ colour changes. In fact, we show that this is the expected number of colour changes for a uniformly random start vertex. This is optimal (up to the constant) when the start vertex is chosen uniformly at random. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypercube geodesics with few colour changes Hollom, Lawrence Combinatorics What is the maximum, over all 2-colourings of the edges of the $n$-dimensional hypercube $Q_n$, of the minimal number of times a path between a vertex $v$ and its antipode $\bar{v}$ changes colour? A conjecture of Norine, in a form due to Feder and Subi, states that this maximum should be 1. The previous best-known upper bound on the number of colour changes was $(\tfrac{5}{16} + o(1))n$ due to Kirchweger, Peitl, Subercaseaux, and Szeider. We improve this bound and answer a question of Leader and Long by finding a geodesic path with at most $(\tfracπ{2} + o(1))\sqrt{n}$ colour changes. In fact, we show that this is the expected number of colour changes for a uniformly random start vertex. This is optimal (up to the constant) when the start vertex is chosen uniformly at random. |
| title | Hypercube geodesics with few colour changes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.20184 |