English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A thoroughly studied problem in Extremal Graph Theory is to find the bestpossible density condition in a host graph $G$ for guaranteeing the presence ofa particular subgraph $H$ in $G$. One such classical result, due toBollob\'{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 onehand, Verstra\"ete conjectured that the quadratic bound $O(k^2)$ wouldguarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision.On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ thereis 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 theedges are replaced by paths of equal length. Recently, Liu and Montgomeryconfirmed 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 abalanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed inThomassen's conjecture and implies the existence of $O(1)$ many vertex-disjointisomorphic $K_k$-subdivisions, confirming Verstra\"ete's conjecture in a strongsense.

Disjoint isomorphic balanced clique subdivisions