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Spectral Turán Problems for Expanded hypergraphs

Zhenyu Ni, Dongquan Cheng, Jing Wang, Liying Kang

Abstract

Given a graph $F$, the expansion $F^{(r)}$ of $F$ is defined as the $r$-uniform hypergraph obtained from $F$ by adding a set of $(r-2)$ distinct new vertices to each edge of $F$. In this paper, we investigate spectral stability results for hypergraphs and their applications.We first establish a spectral stability property: for any $r$-uniform hypergraph containing no copy of the expansion $F^{(r)}$ of a $(k+1)$-chromatic graph $F$, if its $p$-spectral is close to the extremal value, then the hypergraph is structurally close to $T_r(n, k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where sizes of any two parts differ by at most one.Using this spectral stability result, we determine the unique extremal hypergraph that maximizes the $p$-spectral radius among all $n$-vertex $r$-uniform hypergraphs without $t$ vertex-disjoint copies of the expansion $K_{k+1}^{(r)}$ of $K_{k+1}$. We prove that this extremal hypergraph is isomorphic to $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$, the join of the complete $r$-uniform hypergraph $K_{t-1}^{r}$ and $T_r(n-t+1, k)$.As a corollary, we show that $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$ is the unique extremal hypergraph for $tK_{k+1}^{(r)}$, which extends a result of Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] for expanded complete graphs.

Spectral Turán Problems for Expanded hypergraphs

Abstract

Given a graph , the expansion of is defined as the -uniform hypergraph obtained from by adding a set of distinct new vertices to each edge of . In this paper, we investigate spectral stability results for hypergraphs and their applications.We first establish a spectral stability property: for any -uniform hypergraph containing no copy of the expansion of a -chromatic graph , if its -spectral is close to the extremal value, then the hypergraph is structurally close to , the complete -partite -uniform hypergraph on vertices where sizes of any two parts differ by at most one.Using this spectral stability result, we determine the unique extremal hypergraph that maximizes the -spectral radius among all -vertex -uniform hypergraphs without vertex-disjoint copies of the expansion of . We prove that this extremal hypergraph is isomorphic to , the join of the complete -uniform hypergraph and .As a corollary, we show that is the unique extremal hypergraph for , which extends a result of Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] for expanded complete graphs.
Paper Structure (4 sections, 29 theorems, 95 equations)

This paper contains 4 sections, 29 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['F+spectral-stability']}
  4. Proof of Theorem \ref{['tH']}

Key Result

Theorem 1.1

Let $F$ be a graph with $\chi(F)=k+1$, $p>1$, and $k\ge r\ge 3$. For every $\varepsilon>0$, there exist constants $\delta=\delta(k,r,\varepsilon)>0$ and $n_0=n_0(k,r,\varepsilon)$ such that the following holds for all $n>n_0$: If $\mathcal{H}$ is an $n$-vertex $F^{(r)}$-free $r$-graph with then $\mathcal{H}$ is $\varepsilon n^{r}$-close to $T_r(n,k)$, i.e., $\mathcal{H}$ can be transformed to $T_

Theorems & Definitions (47)

  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Proposition 2.1: Nikiforov2014
  • Lemma 2.1: Keevash-Lenz-Mubayi2014
  • Lemma 2.2: CioabaFengTaitZhang2020
  • Lemma 2.3: ChvatalHanson
  • Lemma 2.4: RodlSkokan2006
  • Theorem 2.1: Nikiforov2014
  • ...and 37 more