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| A Criterion for Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Type Theorems and Applications |
| Jianfeng Hou,Xizhi Liu,Hongbin Zhao |
| (Center for Discrete Mathematics, Fuzhou University, Fuzhou 350003,
China;School of Mathematical Sciences, University of Science and Technology
of China, Hefei 230026, China) |
| DOI： |
| Abstract: |
| The classical Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem states that for $\ell\ge 2$, every $n$-vertex $K_{\ell+1}$-free graph with minimum degree greater than $\frac{3\ell-4}{3\ell-1}n$ must be $\ell$-partite.We establish a simple criterion for $r$-graphs, $r \geq 2$, to exhibit an Andr\'{a}sfai--Erd\H{o}s--S\'{o}s 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 Tur\'{a}n 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 $(\pi(F) - \varepsilon_F)\binom{n}{r-1}$ in time $O(n^r)$, where $\varepsilon_F >0$ is a constant.
In particular, for the complete graph $K_{\ell+1}$, we can take $\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$, and this bound is tight up to some multiplicative constant factor unless $\mathbf{W[1]} = \mathbf{FPT}$. 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 $n^{o(\ell)}$ if we replace $\varepsilon_{K_{\ell+1}}$ with $(C\ell)^{-1}$ unless $\mathbf{ETH}$ fails.Furthermore, we apply the degree-stability of $K_{\ell+1}$ to decide the $K_{\ell+1}$-freeness of graphs whose size is close to the Tur\'{a}n bound in time $(\ell+1)n^2$, 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. |
| Key words: Andr´asfai–Erd˝os–S´os Theorem, testing F-freeness, homomorphism, parameterized algorithms, Exponential Time Hypothesis (ETH). |