Bounds on Functionality and Symmetric Difference - Two Intriguing Graph Parameters

Pavel Dvorák, Lukás Folwarczný, Michal Opler, Pavel Pudlák, Robert Sámal, Tung Anh Vu

Abstract: Alecu et al.: Graph functionality, JCTB2021] define functionality, a graph parameter that generalizes graph degeneracy. They research the relation of functionality to many other graph parameters (tree-width, clique-width, VC-dimension, etc.). Extending their research, we prove a logarithmic lower bound for functionality of random graph G(n, p) for large range of p. Previously known graphs have functionality logarithmic in number of vertices. We show that for every graph G on n vertices we have \(\textrm{fun}(G) \le O(\sqrt{ n \log n})\) and we give a nearly matching \(\varOmega (\sqrt{n})\)-lower bound provided by projective planes.