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Domination in Johnson graphs J(n, 3) for odd n
Abstract
In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n,3))$ of the Johnson graph for every even n ≥ 6, expressing it as a closed form ϕn in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd n. We prove this conjecture: γ(J(n,3)) = ϕn for every odd n ≥ 7. A dominating set of J(n,3) is the same as a lotto design LD(n,3,3,2), and the lotto numbers $L(n,3,3,2)$, and hence $γ(J(n,3))$, were already determined for every n by Bate (1978) and by Brouwer (1981). The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.
Related
BibTeX
@misc{lee2026domination,
title = {{Domination in Johnson graphs J(n, 3) for odd n}},
author = {Seung-ah Lee and Semin Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.10326},
eprint = {2606.10326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.10326},
}
RIS
TY - UNPB
TI - Domination in Johnson graphs J(n, 3) for odd n
AU - Seung-ah Lee
AU - Semin Oh
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2606.10326
UR - https://arxiv.org/abs/2606.10326
AB - In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n,3))$ of the Johnson graph for every even n ≥ 6, expressing it as a closed form ϕ_n in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd n. We prove this conjecture: γ(J(n,3)) = ϕ_n for every odd n ≥ 7. A dominating set of J(n,3) is the same as a lotto design LD(n,3,3,2), and the lotto numbers $L(n,3,3,2)$, and hence $γ(J(n,3))$, were already determined for every n by Bate (1978) and by Brouwer (1981). The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.
ER -