Disjoint isomorphic balanced clique subdivisions

TL;DR

This paper proves that a quadratic average degree condition guarantees the existence of balanced and multiple isomorphic subdivisions of complete graphs, advancing extremal graph theory and confirming longstanding conjectures.

Contribution

It establishes the optimal quadratic degree bound for balanced $K_k$-subdivisions and confirms Verstra"ete's conjecture on multiple isomorphic subdivisions.

Findings

Quadratic average degree suffices for balanced $K_k$-subdivisions.

Confirms Verstra"ete's conjecture on multiple isomorphic subdivisions.

Advances understanding of extremal conditions for graph subdivisions.

Abstract

A thoroughly studied problem in Extremal Graph Theory is to find the best possible density condition in a host graph $G$ for guaranteeing the presence of a particular subgraph $H$ in $G$. One such classical result, due to Bollob\'{a}s and Thomason, and independently Koml\'{o}s and Szemer\'{e}di, states that average degree $O(k^2)$ guarantees the existence of a $K_k$-subdivision. We study two directions extending this result. On the one hand, Verstra\"ete conjectured that the quadratic bound $O(k^2)$ would guarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision. On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ there is some $d = d(k)$ such that every graph with average degree at least $d$ contains a balanced subdivision of $K_k$, that is, a copy of $K_k$ where the edges are replaced by paths of equal length. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on $d(k)$ remains open. In this paper, we show that the quadratic bound $O(k^2)$ suffices to force a balanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed in Thomassen's conjecture and implies the existence of $O(1)$ many vertex-disjoint isomorphic $K_k$-subdivisions, confirming Verstra\"ete's conjecture in a strong sense.