Cube edge graph and similar graphs as forbidden structures

Abstract

In extremal graph theory, degenerate (bipartite) extremal graph problems occupy a special place due to Erd{\H{o}}s--Simonovits theorem. One of the smallest graphs for which the asymptotic size of the largest graph avoiding it is unknown is a cube edge graph $Q_8$. Its symmetries and its membership in different classes, such as cycles with all diagonals and prism graphs allows to apply different methods to estimate its extremal numbers. In this thesis a review of known classical and modern methods in degenerate extremal graph theory is given with a focus on the graph $Q_8$ and similar graphs such as hypercube edge graphs and prism graphs. Applicabilities of these approaches are explored and their limitations are analysed.