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Rings are χ-bounded
Abstract
We prove that there is a function f:ℕ→ℕ such that χ(Γ(R))≤ f(ω(Γ(R))) for the zero-divisor graph of any ring R, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming χ(Γ(R))=ω(Γ(R)) for unital commutative rings. While previous counterexamples satisfy χ(Γ(R))≤ ω(Γ(R))+2, we obtain the lower bound f≥ kΩ(log k) among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are χ-bounded.
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BibTeX
@misc{asensio2026rings,
title = {{Rings are $χ$-bounded}},
author = {Sara Asensio and Ignacio García-Marco and Kolja Knauer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17401},
eprint = {2609.17401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17401},
}
RIS
TY - UNPB
TI - Rings are χ-bounded
AU - Sara Asensio
AU - Ignacio García-Marco
AU - Kolja Knauer
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17401
UR - https://arxiv.org/abs/2609.17401
AB - We prove that there is a function f:ℕ→ℕ such that χ(Γ(R))≤ f(ω(Γ(R))) for the zero-divisor graph of any ring R, if the clique number is finite. On the one hand, this consolidates a disproved conjecture of Beck from 1988, claiming χ(Γ(R))=ω(Γ(R)) for unital commutative rings. While previous counterexamples satisfy χ(Γ(R))≤ ω(Γ(R))+2, we obtain the lower bound f≥ k^Ω(log k) among finite commutative rings. Finally, we show that neither zero-divisor graphs of finite commutative semirings nor those of finite commutative nonassociative rings are χ-bounded.
ER -