[Paper Review] On perfect colorings of the halved 24-cube

Denis S. Krotov|ArXiv.org|Mar 2, 2008

graph theory and CDMA systems3 references3 citations

TL;DR

This paper investigates perfect 2-colorings of the even half of the 24-cube graph (H²⁴ₑᵥₑₙ), focusing on colorings with parameter matrix ((20 + c, 256 − c)(c, 276 − c)) corresponding to eigenvalue 20. It proves that such colorings exist for all c from 3 to 128 except c = 1, 2, 4, 5, 7, 10, 13, and shows they do not exist for c = 1, 2, 4, 5, 7. The results are derived using spectral graph theory and combinatorial constructions based on equitable partitions and sphere packing in binary spaces.

ABSTRACT

A vertex 2-coloring of a graph is said to be perfect with parameters $(a_{ij})_{i,j=1}^k$ if for every $i,j\in\{1,...,k\}$ every vertex of color $i$ is adjacent with exactly $a_{ij}$ vertices of color $j$. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube $\{0,1\}^{24}$ with parameters $((20+c,256-c)(c,276-c))$ (i.e., with eigenvalue 20). We prove that such colorings exist for all $c$ from 1 to 128 except 1, 2, 4, 5, 7, 10, 13 and do not exist for $c=1, 2, 4, 5, 7$. Keywords: perfect coloring, equitable partition, hypercube, halved n-cube

Motivation & Objective

• .
• To determine the complete set of admissible parameters c for perfect 2-colorings of the even half of the 24-cube with eigenvalue 20.
• To resolve the existence question for parameters ((20 + c, 256 − c)(c, 276 − c)) by proving existence for most c and non-existence for specific values.
• To establish structural constraints on such colorings using equitable partitions and sphere packing arguments.

Proposed method

• .
• The paper uses spectral graph theory to relate perfect colorings of the 24-cube (distance-1 graph) to those of its distance-2 graph, the halved 24-cube.
• It applies a key lemma showing that a perfect coloring of the hypercube induces a perfect coloring on its distance-2 graph with transformed parameters.
• It constructs explicit colorings using combinatorial designs and sphere packing in the binary space {0,1}^24.
• It proves non-existence for c = 1,2,4,5,7 using contradiction arguments based on counting neighbors and analyzing intersecting spheres.
• It uses a unification lemma to combine disjoint colorings with the same eigenvalue to build new valid colorings.

Experimental results

Research questions

• RQ1 .
• RQ2 For which values of c does a perfect 2-coloring of the halved 24-cube with parameters ((20 + c, 256 − c)(c, 276 − c)) exist?
• RQ3 Why do gaps appear in the spectrum of admissible parameters, particularly for c = 1,2,4,5,7,10,13?
• RQ4 What structural constraints (e.g., on sphere intersections or parity) prevent such colorings from existing for certain c values?

Key findings

• .
• Perfect 2-colorings with parameters ((20 + c, 256 − c)(c, 276 − c)) exist for all c ∈ {3,6,8,9,11,12} ∪ {14,15,…,128}.
• No such colorings exist for c = 1, 2, 4, 5, or 7, as proven by contradiction using neighbor-counting and sphere intersection arguments.
• The values c = 10 and c = 13 remain unresolved, indicating a gap in the spectrum of admissible parameters.
• Colorings with c ≥ 25 or c divisible by 3 are necessary conditions for existence when the coloring is a union of spheres.
• The characteristic function of the coloring is a perfect coloring with eigenvalue 20, linking it to the spectral properties of the halved 24-cube.