Datasets
Open Access
Extended 1-perfect unitrades of length 10

- Citation Author(s):
- Submitted by:
- Denis Krotov
- Last updated:
- Tue, 05/17/2022 - 22:21
- DOI:
- 10.21227/86r8-ry69
- Data Format:
- Link to Paper:
- License:
- Categories:
- Keywords:
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.
In the file "uni10bin.txt", each unitrade C is represented by a line where the words from C are separated by comma.
The file "uni10dec.txt" contains the decimal representation of the same words.
Dataset Files
binary words uni10bin.txt (28.32 kB)
decimal representation uni10dec.txt (10.08 kB)
Open Access dataset files are accessible to all logged in users. Don't have a login? Create a free IEEE account. IEEE Membership is not required.