Balanced Subdivisions of a Large Clique in Graphs with High Average Degree

Abstract

In 1984, Thomassen conjectured that for every constant , there exists such that every graph with average degree at least contains a balanced subdivision of a complete graph on vertices, i.e., a subdivision in which each edge is subdivided the same number of times. Recently, Liu and Montgomery confirmed Thomassen’s conjecture. We show that for every constant , every graph with average degree at least contains a balanced subdivision of a complete graph of size at least . Note that this bound is almost optimal. Moreover, we show that every sparse expander with minimum degree at least contains a balanced subdivision of a complete graph of size at least .