A finiteness theorem for geodesic Leech wheels
Abstract
Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.
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BibTeX
@misc{yim2026finiteness,
title = {{A finiteness theorem for geodesic Leech wheels}},
author = {Junyeop Yim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02544},
eprint = {2609.02544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02544},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A finiteness theorem for geodesic Leech wheels
AU - Junyeop Yim
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02544
UR - https://arxiv.org/abs/2609.02544
AB - Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.
ER -