Extended 1-perfect unitrades of length 10

Citation Author(s):
Denis
Krotov
Sobolev Institute of Mathematics
Submitted by:
Denis Krotov
Last updated:
Tue, 05/17/2022 - 22:21
DOI:
10.21227/86r8-ry69
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Abstract 

The weight of a binary word is the number of ones in it. The N-cube is the graph whose vertices are the binary words of length N and the edges are the pairs of words differing in exactly one position. The halved N-cube is the graph whose vertices are the even-weight binary words of length N and the edjes are the pairs of words differing in exactly two positions. An extended 1-perfect unitrade is a set C of binary even-weight words of length N such that every odd-weight word is adjacent to exactly 0 or 2 words from C. Equivalently, an extended 1-perfect unitrade is a set of vertices of the halved N-cube that induces a N/2-regular subgraph without triangles. An extended 1-perfect unitrade is primary if it is non-empty and cannot be split into two non-empty extended 1-perfect unitrades.
The dataset is the result of the exhaustive search of primary extended 1-perfect unitrades in the 10-cube.
We list representatives of the all 38 equivalence classes (two sets A and B of binary N-words are equivlent if there is an automorphism of the N-cube that sends A to B).

The unitrades from the first 8 classes are bibartite and were described in [D.S.Krotov, The extended 1-perfect trades in small hypercubes. Discrete Math. 340(10) 2017, 2559-2572. Section 5] as extended 1-perfect bitrades.

The representatives of the equivalence classes No 0, 1, 2, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20 are constant-weight; the other 23 classes have no constant-weight representatives. The unitrades from classes No 0, 1, 2, 4, 5, 8, 9 are reducible to unitrades of length 8.

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