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On generalized Turán problems with bounded matching number

Yisai Xue, Liying Kang

Abstract

The generalized Turán number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of Turán problems with a bounded matching number.In this paper, we establish stability results for generalized Turán problems with bounded matching number.Using the stability results, we provide exact values of $\ex(n,K_r,\{F,M_{s+1}\})$ for $F$ being any non-bipartite graph or a path on $k$ vertices.

On generalized Turán problems with bounded matching number

Abstract

The generalized Turán number is defined as the maximum number of copies of a graph in an -vertex graph that does not contain any graph . Alon and Frankl initiated the study of Turán problems with a bounded matching number.In this paper, we establish stability results for generalized Turán problems with bounded matching number.Using the stability results, we provide exact values of for being any non-bipartite graph or a path on vertices.

Paper Structure

This paper contains 7 sections, 15 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Forbidding non-bipartite graphs
  4. Forbidding bipartite graphs
  5. Stability result
  6. Forbidding even paths
  7. Forbidding odd paths

Key Result

Theorem 1.1

Let $r\geq 3$ and $z>0$ be constants. Suppose $s$ is an integer and $F$ is a graph with $\chi(F)\ge r$. Assume there exists $n_0=n_0(s,r,z)$ such that for every graph $G$ on $n>n_0$ vertices satisfies that then the following holds.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.3
  • Remark 1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 2
  • Theorem 2.1: Tutte-Berge Theorem lovasz2009matching
  • Proposition 2.2: alon2016many
  • ...and 41 more