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A new view of hypercube genus

Richard H. Hammack, Paul C. Kainen

TL;DR

The paper presents a new, visual proof that the n-cube $Q_n$ embeds on an orientable surface of genus $\gamma(Q_n)=1+(n-4)2^{n-3}$ by constructing a genus surface $T$ from square faces of the $n$-cube's $2$-skeleton. This surface $T$ contains a embedding of $Q_n$, and orientability is established, yielding the exact genus via Euler’s formula $\\gamma(T)=(2-v+e-f)/2$, with a matching lower bound derived for bipartite graphs. For odd $n$, the authors further show the $2$-skeleton decomposes into $(n-1)/2$ isometric copies of the genus surface that intersect pairwise in $Q_n$, forming a parallel family. They also connect these constructions to Hamiltonian decompositions of $K_n$ and discuss generalizations and related skew-polyhedron perspectives, offering a cohesive intrinsic viewpoint on hypercube genus with potential extensions to nonorientable cases.

Abstract

Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the $n$-dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube's 2-skeleton. For odd dimension $n$, the entire 2-skeleton decomposes into $(n-1)/2$ copies of the surface, and the intersection of any two copies is the hypercube graph.

A new view of hypercube genus

TL;DR

The paper presents a new, visual proof that the n-cube embeds on an orientable surface of genus by constructing a genus surface from square faces of the -cube's -skeleton. This surface contains a embedding of , and orientability is established, yielding the exact genus via Euler’s formula , with a matching lower bound derived for bipartite graphs. For odd , the authors further show the -skeleton decomposes into isometric copies of the genus surface that intersect pairwise in , forming a parallel family. They also connect these constructions to Hamiltonian decompositions of and discuss generalizations and related skew-polyhedron perspectives, offering a cohesive intrinsic viewpoint on hypercube genus with potential extensions to nonorientable cases.

Abstract

Beineke, Harary and Ringel discovered a formula for the minimum genus of a torus in which the -dimensional hypercube graph can be embedded. We give a new proof of the formula by building this surface as a union of certain faces in the hypercube's 2-skeleton. For odd dimension , the entire 2-skeleton decomposes into copies of the surface, and the intersection of any two copies is the hypercube graph.
Paper Structure (5 sections, 3 theorems, 6 equations, 6 figures)

This paper contains 5 sections, 3 theorems, 6 equations, 6 figures.

Table of Contents

  1. introduction.
  2. Hypercubes.
  3. Two-cell embeddings of graphs.
  4. Genus embeddings of hypercubes.
  5. Parallel genus embeddings.

Key Result

Lemma 1

If $G$ has $v$ vertices, $e$ edges, and is bipartite, then ${\gamma(G)\geq\frac{1}{4}(4-2v+e)}$.

Figures (6)

  • Figure 1: Left: The complete bipartite graph $K_{m,n}$ can be regarded as having $m$ black vertices, $n$ white vertices, and an edge joining any two vertices of different colors. Right: $K_{3,3}$, $K_{4,4}$, and $K_{4,5}$ on surfaces.
  • Figure 2: The 2-, 3-, and 4-dimensional cubes.
  • Figure 3: Examples of Tori. The sphere $T_0$ (left) followed by $T_1$, $T_2$, and $T_3$.
  • Figure 4: The squares in $\mathcal{F}$ that surround a vertex.
  • Figure 5: Left: A Möbius strip in $H_4$. Right: Squares in $T$ are oriented by the right-hand rule at black vertices. Squares bicolored $(i{-}1)i$, $i(i{+}1)$, and $(i{+}1)(i{+}2)$ are shown.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof