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arxiv: 2605.20184 · v2 · pith:ZJ4RP26Anew · submitted 2026-05-19 · 🧮 math.CO

Hypercube geodesics with few colour changes

Lawrence Hollom This is my paper

Pith reviewed 2026-05-25 06:00 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C1505C38
keywords hypercubeedge coloringgeodesicscolor changesantipodal pathsprobabilistic method
1
0 comments X

The pith

In any 2-edge-coloring of the n-dimensional hypercube, some shortest path from a vertex to its antipode changes color at most (π/2 + o(1))√n times.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for every 2-coloring of the edges of Q_n there exists a geodesic from any chosen vertex v to its antipode that switches colors no more than (π/2 + o(1))√n times. The same quantity is shown to be the exact expected number of color changes when the starting vertex is drawn uniformly at random. This improves the earlier linear upper bound of (5/16 + o(1))n and supplies the answer to a question of Leader and Long. The argument proceeds by first computing the expectation over random geodesics and then deducing the worst-case bound from it.

Core claim

For any 2-coloring of the edges of Q_n the minimum number of color changes on a geodesic from v to its antipode is at most (π/2 + o(1))√n; moreover this quantity equals the expected number of color changes when v is chosen uniformly at random, and the bound is asymptotically tight for random choice of v.

What carries the argument

The expected number of color changes along a uniformly random geodesic from a uniformly random starting vertex to its antipode.

If this is right

  • The previous upper bound of (5/16 + o(1))n is replaced by a sublinear bound of order √n.
  • The result gives the precise asymptotic order of the expected number of color changes for a random starting vertex.
  • The conjecture that a single color change always suffices remains open, but any counter-example must require at least Ω(√n) changes on average.
  • The same expectation calculation immediately yields an explicit construction of a low-change geodesic for any fixed coloring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the constant π/2 cannot be lowered, then √n is the correct growth rate rather than a constant.
  • The probabilistic averaging technique may adapt to edge colorings with more than two colors or to other highly symmetric graphs.
  • Explicit computation of the expectation for small n would reveal how fast the o(1) term approaches zero.

Load-bearing premise

The worst-case number of color changes over all colorings is bounded above by the average number of color changes taken over a random starting vertex.

What would settle it

Produce a 2-edge-coloring of Q_n together with a vertex v such that every geodesic from v to its antipode changes color more than (π/2 + ε)√n times for some fixed ε > 0.

Figures

Figures reproduced from arXiv: 2605.20184 by Lawrence Hollom.

Figure 1
Figure 1. Figure 1: A pictorial representation of the functions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

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 $(\tfrac{\pi}{2} + o(1))\sqrt{n}$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript considers the maximum, over all 2-edge-colorings of the hypercube Q_n, of the minimum number of color changes along any geodesic from a vertex v to its antipode. It improves the prior upper bound of (5/16 + o(1))n to (π/2 + o(1))√n by establishing that, for every coloring, the expected number of color changes over a uniform random starting vertex is at most (π/2 + o(1))√n; the desired uniform bound then follows by averaging. The same expectation is shown to be asymptotically tight when the starting vertex is chosen uniformly at random, answering a question of Leader and Long.

Significance. If the central probabilistic argument holds, the result supplies a qualitatively stronger bound (square-root rather than linear in n) and resolves an explicit open question. The approach via direct expectation over random vertices is clean and yields both the upper bound on the worst-case coloring and the matching lower bound for the average case.

minor comments (2)
  1. The abstract states that the expectation yields the uniform upper bound; a brief sentence in §1 or §2 confirming that this is a direct averaging argument (min ≤ E) would make the logical step explicit for readers.
  2. The o(1) term in the main bound should be accompanied by a short statement of the rate (e.g., O(1/√n) or similar) once the proof is complete, to clarify the error term.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation relies on a standard probabilistic averaging argument: an upper bound on the expected number of color changes over a uniform random starting vertex v directly implies the same upper bound on the minimum over v (hence on the maximum over colorings of that minimum). This is a direct consequence of the definition of expectation and does not reduce any claimed result to a fitted parameter, self-definition, or self-citation chain. No equations or steps in the provided text exhibit the enumerated circularity patterns; the result is presented as obtained from new arguments independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies only on the standard definition of the hypercube graph and the notion of 2-edge-colorings; no free parameters, new axioms, or invented entities are introduced or fitted.

axioms (2)
  • standard math The n-dimensional hypercube Q_n has vertices as binary strings of length n and edges corresponding to single bit flips.
    Standard definition invoked in the problem statement.
  • standard math A geodesic from v to its antipode has length exactly n.
    Basic property of the hypercube used to define the paths under consideration.

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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