Graphs with Large Girth and Chromatic Number are Hard for Nullstellensatz
Abstract.
We study the computational efficiency of approaches, based on Hilbert’s Nullstellensatz, which use systems of linear equations for detecting noncolorability of graphs having large girth and chromatic number. We show that for every non-\(k\)-colorable graph with \(n\) vertices and girth \(g\gt 4k\), the algorithm is required to solve systems of size at least \(n^{\Omega (g)}\) in order to detect its non-\(k\)-colorability.
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Acknowledgments.
The authors thank Jesús A. De Loera for valuable comments, insights, and feedback during this research, in particular, for bringing [24] to our attention. We would like also to thank the anonymous reviewers for their comments and suggestions. Finally, the authors thank Bertrand Guenin and Jochen Koenemann for their valuable comments on a preliminary version of the results.
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Information & Authors
Information
Published In
SIAM Journal on Discrete Mathematics
Pages: 2108 - 2131
DOI: 10.1137/23M1553273
ISSN (online): 1095-7146
Copyright
© 2024 Julian Romero and Levent Tunçel.
History
Submitted: 14 February 2023
Accepted: 28 March 2024
Published online: 11 July 2024
Keywords
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Authors
Funding Information
Tutte Scholarship
Office of Naval Research (ONR): N00014-15-1-2171, N00014-18-1-2078
Funding: This work was funded in part by NSERC Discovery Grants, Tutte Scholarship, U.S. Office of Naval Research under award numbers: N00014-15-1-2171 and N00014-18-1-2078. This financial support is gratefully acknowledged.
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