Phase Transitions of Structured Codes of Graphs

07/17/2023

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by   Bo Bai, et al.

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We consider the symmetric difference of two graphs on the same vertex set [n], which is the graph on [n] whose edge set consists of all edges that belong to exactly one of the two graphs. Let ℱ be a class of graphs, and let M_ℱ(n) denote the maximum possible cardinality of a family 𝒢 of graphs on [n] such that the symmetric difference of any two members in 𝒢 belongs to ℱ. These concepts are recently investigated by Alon, Gujgiczer, Körner, Milojević, and Simonyi, with the aim of providing a new graphic approach to coding theory. In particular, M_ℱ(n) denotes the maximum possible size of this code. Existing results show that as the graph class ℱ changes, M_ℱ(n) can vary from n to 2^(1+o(1))n2. We study several phase transition problems related to M_ℱ(n) in general settings and present a partial solution to a recent problem posed by Alon et. al.