Balanced subdivisions of cliques in graphs

TL;DR

This paper proves optimal bounds for the existence of balanced clique subdivisions in graphs with high average degree, confirming longstanding conjectures and extending results to specific graph classes.

Contribution

It establishes near-optimal degree conditions for balanced clique subdivisions, confirming conjectures and extending previous results to $C_4$-free graphs.

Findings

Graphs with high average degree contain large balanced clique subdivisions.

Confirmed that high average degree guarantees disjoint isomorphic subdivisions of $K_d$.

Extended results to $C_4$-free graphs with large average degree.

Abstract

Given a graph $H$, a balanced subdivision of $H$ is a graph obtained from $H$ by subdividing every edge the same number of times. In 1984, Thomassen conjectured that for each integer $k\ge 1$, high average degree is sufficient to guarantee a balanced subdivision of $K_k$. Recently, Liu and Montgomery resolved this conjecture. We give an optimal estimate up to an absolute constant factor by showing that there exists $c>0$ such that for sufficiently large $d$, every graph with average degree at least $d$ contains a balanced subdivision of a clique with at least $cd^{1/2}$ vertices. It also confirms a conjecture from Verstra{\"e}te: every graph of average degree $cd^2$, for some absolute constant $c>0$, contains a pair of disjoint isomorphic subdivisions of the complete graph $K_d$. We also prove that there exists some absolute $c>0$ such that for sufficiently large $d$, every $C_4$-free graph with average degree at least $d$ contains a balanced subdivision of the complete graph $K_{cd}$, which extends a result of Balogh, Liu and Sharifzadeh.