On perfect colorings of the halved 24-cube

Details On perfect colorings of the halved 24-cube On perfect colorings of the halved 24-cube

2008-03-02

arxiv.org/abs/0803.0068v2

Diskretn. Anal. Issled. Oper. 15(5) 2008, 35-46 (in Russian)

Mathematics

Combinatorics

Eng: 9p; Rus: 10p. V.2: case c=7 added; title changed; minor revision

Scientific paper

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

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