The Rainbow Saturation Number Is Linear
Abstract.
Given a graph \(H\), we say that an edge-colored graph \(G\) is \(H\)-rainbow saturated if it does not contain a rainbow copy of \(H\), but the addition of any nonedge in any color creates a rainbow copy of \(H\). The rainbow saturation number \(\mathrm{rsat}(n,H)\) is the minimum number of edges among all \(H\)-rainbow saturated edge-colored graphs on \(n\) vertices. We prove that for any nonempty graph \(H\), the rainbow saturation number is linear in \(n\), thus proving a conjecture of Girão, Lewis, and Popielarz. In addition, we give an improved upper bound on the rainbow saturation number of the complete graph, disproving a second conjecture of Girão, Lewis, and Popielarz.
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Acknowledgment.
We would like to thank the anonymous referees for their helpful comments.
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Information & Authors
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Published In
SIAM Journal on Discrete Mathematics
Pages: 1239 - 1249
DOI: 10.1137/23M1566881
ISSN (online): 1095-7146
Copyright
© 2024 the authors.
History
Submitted: 19 April 2023
Accepted: 21 December 2023
Published online: 8 April 2024
Keywords
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Funding Information
Natural Sciences and Engineering Research Council of Canada (NSERC): RGPIN-2021-02511, DGECR-2021-00047
Funding: The first author’s research was supported by a PIMS postdoctoral fellowship. The third author’s research was supported by the Royal Society. The fourth author’s research was supported by NSERC Discovery Grant RGPIN-2021-02511 and NSERC Early Career Supplement DGECR-2021-00047. The fifth author was supported by NSERC CGS M.
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- Weak Rainbow Saturation Numbers of GraphsJournal of Graph Theory, Vol. 109, No. 1 | 6 January 2025
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