Date of Award

5-2025

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematics

Thesis Sponsor/Dissertation Chair/Project Chair

Deepak Bal

Committee Member

Jonathan Cutler

Committee Member

Ashwin Vaidya

Abstract

The concept of graph codes, introduced in recent work by Alon, applies coding theory to graphs by defining codewords as graphs and measuring distances between them through structural differences. In ordinary coding theory, the distance between two words is the number of positions in which the two words differ. In graph codes, each word is a graph and the distance between two graphs is measured by structural properties of their difference. For a fixed graph H, an H-code is a collection F of graphs on vertex set [n] = {1, 2, . . . , n} such that the symmetric difference of any two graphs in F is not a copy of H. Alon conjectured that whenever H has an even number of edges, any H-code must be a negligible fraction of the set of all graphs on [n]. He proved this conjecture for all H of a certain type. In this thesis, we extend the notion of graph codes to hypergraphs and directed graphs and prove analogs of Alon’s result in these settings.

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Mathematics Commons

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