Embedding clique subdivisions via crux

Abstract

For a graph $G$ and a constant $\hat{I}\pm >0$, we denote by $C_{\hat{I}}\pm (G)$ the minimum order of a subgraph $H\subseteq G$ with $d(H)\ge \hat{I}\pm d(G)$. Liu and Montgomery conjectured that every graph $G$ contains $K_{\hat{I}\text{c◯}(t)}$ as a subdivision for $t=\min \{d(G),\sqrt{\frac{C_{\hat{I}}\pm (G)}{\log C_{\hat{I}}\pm (G)}}\}$. In the paper, we prove this conjecture.