Graduation Term

Spring 2026

Degree Name

Master of Science (MS)

Department

Department of Mathematics

Committee Chair

Amin Bahmanian

Committee Member

Sunil Chebolu

Committee Member

Papa Sissokho

Committee Member

Charlotte Ure

Abstract

Let [n] = {1, ..., n}. A hypercube H of order n and dimension d is a d-dimensional array whose nᵈ cells are indexed by [n]ᵈ. A hyperplane in H is obtained by fixing one coordinate, while allowing the remaining d–1 coordinates to vary. We wish to color each cell of H from a palette of nd-1 colors such that each hyperplane is polychromatic.

Our main result is the following. Let n be sufficiently large. There exists a symmetric coloring of the d-dimensional hypercubes of order n whose all hyperplanes are polychromatic if and only if:

n ≢ s (mod d) where 0 < s ≤ d/2

We show that a d-dimensional symmetric layer-rainbow Latin hypercube of order n is equivalent to a 1-factorization of ∪ᵢ₌₁ᵈ (i–1)! S(d,i) Kₙⁱ. Using a network flow argument and a generalization of Baranyai's theorem, we reduce the problem to solving the system of equations:

  • a1,i + 2a2,i + ... + d ad,i = n for i ∈ {1, ..., nᵈ⁻¹}

  • Σi=1(nᵈ⁻¹) ak,i = S(d,k)(k–1)! C(n,k) for k ∈ {1, ..., d}

The case d = 2 corresponds to very well-known objects known as symmetric Latin squares.

Access Type

Thesis-Open Access

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