Balanced subdivisions of a large clique in graphs with high average degree

TL;DR

This paper proves that graphs with high average degree contain large balanced subdivisions of complete graphs, nearly matching the known bounds, and extends results to sparse expanders with high minimum degree.

Contribution

It establishes nearly optimal bounds for the size of balanced subdivisions in graphs with high average degree and extends to sparse expanders with high minimum degree.

Findings

Graphs with average degree at least d contain balanced K_k subdivisions with k = Ω(d^c) for 0 < c < 1/2.

Sparse expanders with minimum degree at least d contain balanced K_k subdivisions with k = Ω(d).

Bounds are nearly optimal, advancing understanding of graph subdivisions.

Abstract

In 1984, Thomassen conjectured that for every constant $k \in \mathbb{N}$, there exists $d$ such that every graph with average degree at least $d$ contains a balanced subdivision of a complete graph on $k$ 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 $0<c<1/2$, every graph with average degree at least $d$ contains a balanced subdivision of a complete graph of size at least $\Omega(d^{c})$. Note that this bound is almost optimal. Moreover, we show that every sparse expander with minimum degree at least $d$ contains a balanced subdivision of a complete graph of size at least $\Omega(d)$.