Year: 2026
Type: article
Abstract: The classical Andrásfai–Erdős–Sόs Theorem states that for $ℓ≥2$, every $n$-vertex $K_{ℓ+1}$-free graph with minimum degree greater than $\frac{3ℓ-4}{3ℓ-1}n$ must be $ℓ$-partite. We establish a simple criterion for $r$-graphs, $r≥2$, to exhibit an Andrásfai–Erdős–Sόs type property, also known as degree-stability. This leads to a classification of most previously studied hypergraph families with this property. An immediate application of this result, combined with a general theorem by Keevash–Lenz–Mubayi, solves the spectral Turán problems for a large class of hypergraphs. We show an interesting application of the degree-stability in Complexity Theory: For every $r$-graph $F$ with degree-stability, there is a simple algorithm to decide the $F$-freeness of an $n$-vertex $r$-graph with minimum degree greater than $(π(F)-ε_F)\binom{n}{r-1}$ in time $O(n^r)$, where $ε_F>0$ is a constant. In particular, for the complete graph $K_{ℓ+1}$, we can take $ε_{K_{ℓ+1}}=(3ℓ^2-ℓ)^{-1}$, and this bound is tight up to some multiplicative constant factor unless W[1]=FPT. Based on a result by Chen–Huang–Kanj–Xia, we further show that for every fixed $C>0$, this problem cannot be solved in time $n^{o(ℓ)}$ if we replace $ε_{K_{ℓ+1}}$ with $(Cℓ)^{-1}$ unless ETH fails. Furthermore, we apply the degree-stability of $K_{ℓ+1}$ to decide the $K_{ℓ+1}$-freeness of graphs whose size is close to the Turán bound in time $(ℓ+1)n^2$, partially improving a recent result by Fomin–Golovach–Sagunov–Simonov. As an intermediate step, we show that for a specific class of $r$-graphs $F$, the (surjective) $F$-coloring problem can be solved in time $O(n^r)$, provided the input $r$-graph has $n$ vertices and a large minimum degree, refining several previous results.
Source: Annals of Applied Mathematics
Authors: Jianfeng Hou, Xizhi Liu, Hongbin Zhao
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W7163638108