Graph decompositions and related extremal problems

Abstract

We present several results on graph decompositions and related extremal problems. The first result discusses the decomposition of graphs with no odd complete minor, the second result investigates how minor-monotone changes after adding a few random edges, and how the structure of a given graph changes after these random edge perturbations. The third and fourth results focus on directed graphs and their extremal properties, and study how many edges can be deleted while preserving the vertex/edge-connectivity of a given highly connected tournament-like digraph, or how a highly connected tournament-like digraph can be partitioned into many highly connected pieces with desired shapes. The last result is related to the rational exponent conjecture by Erdős and Simonovits in 1980s, which studies how many edges an $n$-vertex graph $G$ can have if $G$ does not contain a bipartite graph $H$ as a subgraph, and the exponents of these extremal functions.