Structured Codes of Graphs
Abstract.
We investigate the maximum size of graph families on a common vertex set of cardinality \(n\) such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of \(n\) when the prescribed condition is connectivity or 2-connectivity, Hamiltonicity, or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.
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Acknowledgment.
We thank Gábor Tardos for his remark presented as Remark 3.4.
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© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 31 March 2022
Accepted: 29 September 2022
Published online: 3 March 2023
Keywords
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Funding Information
National Science Foundation (NSF): DMS-1855464
Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (NKFI): K-120706, K-132696, SNN-135643
Funding: The first author’s research was supported in part by NSF grant DMS-1855464 and BSF grant 2018267. The second author’s research was partially supported by the National Research, Development and Innovation Office (NKFIH) grant K-120706 of NKFIH Hungary. The fifth author’s research was partially supported by the National Research, Development and Innovation Office (NKFIH) grants K-120706, K-132696, and SNN-135643 of NKFIH Hungary.
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