Spanning clique subdivisions in pseudorandom graphs
Hyunwoo Lee, Matías Pavez-Signé, Teo Petrov
TL;DR
The paper investigates when a host graph with pseudorandom structure must contain a spanning subdivision of a clique $K_t$. It advances three regimes: (i) spectral-gap regime with $d/\lambda \ge C$ yielding spanning $K_t$-subdivisions up to $t \le \min\{cd, c\sqrt{n/\log n}\}$; (ii) stronger polylogarithmic gap $d/\lambda \ge C\log^3 n$ producing spanning nearly-balanced subdivisions up to $t \le \min\{cd, c\sqrt{n/\log^3 n}\}$; (iii) linear minimum degree plus no large bipartite holes giving spanning nearly-balanced subdivisions for $t \le c\sqrt{n}$. The authors develop and combine three core tools: the extendability method to incrementally build large connected structures, sorting networks in sparse expanders to balance subdivision paths, and the absorption technique to achieve spanning results. These techniques yield Dirac-type spanning subdivision results and imply that randomly perturbed graphs inherit spanning nearly-balanced $K_t$-subdivisions with high probability. Overall, the work broadens our understanding of how pseudorandomness guarantees spanning complex substructures and connects to random perturbations of dense graphs.
Abstract
In this paper, we study the appearance of a spanning subdivision of a clique in graphs satisfying certain pseudorandom conditions. Specifically, we show the following three results. Firstly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/λ\ge C$, every $(n,d,λ)$-graph contains a spanning subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log n}}\}$. Secondly, that there are constants $C>0$ and $c\in (0,1]$ such that, whenever $d/λ\ge C\log^3n$, every $(n,d,λ)$-graph contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le \min\{cd,c\sqrt{\frac{n}{\log^3n}}\}$. Finally, we show that for every $μ>0$, there are constants $c,\varepsilon\in (0,1]$ and $n_0\in \mathbb N$ such that, whenever $n\ge n_0$, every $n$-vertex graph with minimum degree at least $μn$ and no bipartite holes of size $\varepsilon n$ contains a spanning nearly-balanced subdivision of $K_t$ for all $2\le t \le c\sqrt{n}$.