Phase Transitions of Structured Codes of Graphs

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 2, Page 1902-1914, June 2024.
Abstract. We consider the symmetric difference of two graphs on the same vertex set [math], which is the graph on [math] whose edge set consists of all edges that belong to exactly one of the two graphs. Let [math] be a class of graphs, and let [math] denote the maximum possible cardinality of a family [math] of graphs on [math] such that the symmetric difference of any two members in [math] belongs to [math]. These concepts have been recently investigated by Alon et al. [SIAM J. Discrete Math., 37 (2023), pp. 379–403] with the aim of providing a new graphic approach to coding theory. In particular, [math] denotes the maximum possible size of this code. Existing results show that as the graph class [math] changes, [math] can vary from [math] to [math]. We study several phase transition problems related to [math] in general settings and present a partial solution to a recent problem posed by Alon et al.