Zero-Sum Cycles in Regular Digraphs

[2608.14515] Zero-Sum Cycles in Regular Digraphs Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14515 (math) [Submitted on 14 Aug 2026] Title: Zero-Sum Cycles in Regular Digraphs Authors: Varun Sivashankar View a PDF of the paper titled Zero-Sum Cycles in Regular Digraphs, by Varun Sivashankar View PDF HTML (experimental) Abstract: Let $\Gamma$ be a finite group of order $k\ge2$, and label the edges of a simple loopless $d$-regular digraph $D$ by elements of $\Gamma$. A directed cycle is zero-sum if the ordered product of its labels is the identity of $\Gamma$. We prove that a zero-sum cycle exists whenever $d\ge e^3(k-1)$. We also prove that every labelled $d$-regular digraph contains $\Omega(d/k)$ pairwise vertex-disjoint zero-sum cycles. When $d\ge50k$, it contains $\Omega(d^2/k)$ pairwise edge-disjoint zero-sum cycles. All three results are asymptotically optimal. The existence and packing results extend to Eulerian digraphs whose minimum and maximum common degrees $\delta$ and $\Delta$ satisfy $\delta^3/\Delta^2=\Omega(k)$. The techniques extend a determinant--permanent argument of Friedland for even directed cycles. Subjects: Combinatorics (math.CO) MSC classes: 05C35 Cite as: arXiv:2608.14515 [math.CO] (or arXiv:2608.14515v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14515 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Varun Sivashankar [ view email ] [v1] Fri, 14 Aug 2026 17:36:32 UTC (23 KB) Full-text links: Access Paper: View a PDF of the paper titled Zero-Sum Cycles in Regular Digraphs, by Varun Sivashankar View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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