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On generalized Tur{á}n problems with bounded matching number and circumference

Yongchun Lu, Liying Kang, Yisai Xue

TL;DR

This work determines the exact generalized Turán numbers ex(n, K_r, {C_{≥k}, M_{s+1}}) for all s, r, k in the large-n regime. The authors deploy a stability framework anchored by a canonical partition and a suite of structural lemmas to reduce extremal configurations to explicit constructions. They distinguish cases by the parity of k and by residual decompositions of s relative to p, yielding precise formulas and corresponding extremal graphs G_1–G_6. The results extend the understanding of Turán-type problems with circumference constraints and bounded matching numbers, with constructions achieving tight bounds and clear combinatorial structure.

Abstract

Let \( \mathcal{F} \) be a family of graphs. The generalized Turán number \( \operatorname{ex}(n, K_r, \mathcal{F}) \) is the maximum number of $K_r$ in an \( n \)-vertex graph that does not contain any member of \( \mathcal{F} \) as a subgraph. Recently, Alon and Frankl initiated the study of Turán problems with bounded matching number. In this paper, we determine the generalized Turán number of \( C_{\geq k} \) with bounded matching number.

On generalized Tur{á}n problems with bounded matching number and circumference

TL;DR

This work determines the exact generalized Turán numbers ex(n, K_r, {C_{≥k}, M_{s+1}}) for all s, r, k in the large-n regime. The authors deploy a stability framework anchored by a canonical partition and a suite of structural lemmas to reduce extremal configurations to explicit constructions. They distinguish cases by the parity of k and by residual decompositions of s relative to p, yielding precise formulas and corresponding extremal graphs G_1–G_6. The results extend the understanding of Turán-type problems with circumference constraints and bounded matching numbers, with constructions achieving tight bounds and clear combinatorial structure.

Abstract

Let be a family of graphs. The generalized Turán number \( \operatorname{ex}(n, K_r, \mathcal{F}) \) is the maximum number of in an -vertex graph that does not contain any member of as a subgraph. Recently, Alon and Frankl initiated the study of Turán problems with bounded matching number. In this paper, we determine the generalized Turán number of with bounded matching number.

Paper Structure

This paper contains 4 sections, 12 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Forbidding $C_{\geq 2p-1}$
  4. Forbidding $C_{\geq 2p}$

Key Result

Theorem 1.1

For any $s \geq 2$ and $n \geq 2 k+1$, we have

Theorems & Definitions (59)

  • Theorem 1.1: wang2020shifting
  • Theorem 1.2: alon2024turan
  • Theorem 1.3: gerbner2024turan
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1: dirac1952some
  • Lemma 2.2: kopylov1977maximal
  • Lemma 2.3: chakraborti2021many
  • Lemma 2.4
  • proof
  • ...and 49 more