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Induced Turán problem in bipartite graphs

Maria Axenovich, Jakob Zimmermann

Abstract

The classical extremal function for a graph $H$, $ex(K_n, H)$ is the largest number of edges in a subgraph of $K_n$ that contains no subgraph isomorphic to $H$. Note that defining $ex(K_n, H-ind)$ by forbidding induced subgraphs isomorphic to $H$ is not very meaningful for a non-complete $H$ since one can avoid it by considering a clique. For graphs $F$ and $H$, let $ex(K_n, \{F, H-ind\})$ be the largest number of edges in an $n$-vertex graph that contains no subgraph isomorphic to $F$ and no induced subgraph isomorphic to $H$. Determining this function asymptotically reduces to finding either $ex(K_n, F)$ or $ex(K_n, H)$, unless $H$ is a biclique or both $F$ and $H$ are bipartite. Here, we consider the bipartite setting, $ex(K_{n,n}, \{F, H-ind\})$ when $K_n$ is replaced with $K_{n,n}$, $F$ is a biclique, and $H$ is a bipartite graph. Our main result, a strengthening of a result by Sudakov and Tomon, implies that for any $d\geq 2$ and any $K_{d,d}$-free bipartite graph $H$ with each vertex in one part of degree either at most $d$ or a full degree, so that there are at most $d-2$ full degree vertices in that part, one has $ex(K_{n,n}, \{K_{t,t}, H-ind\}) = o(n^{2-1/d})$. This provides an upper bound on the induced Turán number for a wide class of bipartite graphs and implies in particular an extremal result for bipartite graphs of bounded VC-dimension by Janzer and Pohoata.

Induced Turán problem in bipartite graphs

Abstract

The classical extremal function for a graph , is the largest number of edges in a subgraph of that contains no subgraph isomorphic to . Note that defining by forbidding induced subgraphs isomorphic to is not very meaningful for a non-complete since one can avoid it by considering a clique. For graphs and , let be the largest number of edges in an -vertex graph that contains no subgraph isomorphic to and no induced subgraph isomorphic to . Determining this function asymptotically reduces to finding either or , unless is a biclique or both and are bipartite. Here, we consider the bipartite setting, when is replaced with , is a biclique, and is a bipartite graph. Our main result, a strengthening of a result by Sudakov and Tomon, implies that for any and any -free bipartite graph with each vertex in one part of degree either at most or a full degree, so that there are at most full degree vertices in that part, one has . This provides an upper bound on the induced Turán number for a wide class of bipartite graphs and implies in particular an extremal result for bipartite graphs of bounded VC-dimension by Janzer and Pohoata.
Paper Structure (4 sections, 10 theorems, 14 equations, 1 figure)

This paper contains 4 sections, 10 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations, Definitions, and Preliminary Lemmas
  3. Proofs of the main results
  4. Concluding remarks

Key Result

Theorem 1

Let $k, r, d, t$ be fixed non-negative integers, $r+2 \leq d \leq k$, and $\epsilon$ be an arbitrary positive number. Then there is $n_0$ such that for any $n>n_0$, if $G=(A, B; E)$ is a bipartite graph, $|A|=|B|=n$ and $||G|| \geq \epsilon n^{2-\frac{1}{d}}$, then $G$ contains either a copy of $K_{

Figures (1)

  • Figure 1: A graph $W(k, d, r)$ with $r=2$ and $d=6$.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2: Sudakov and Tomon sudakov2019turan
  • Theorem 3: Janzer and Pohoata janzer2021zarankiewicz
  • Lemma 1
  • proof
  • Lemma 2: Hypergraph Removal Lemma
  • Lemma 3: Kővari-Sós-Turán Theorem, KST
  • Lemma 4: Reduction Lemma
  • Lemma 5: Deletion-method Lemma
  • Lemma 6
  • ...and 6 more