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Bipartite Turán problem on cographs

Jakob Paul Zimmermann

TL;DR

This work analyzes ex$(n, \{K_{s,t}, P_4\text{-ind}\})$ within the restricted framework of cographs, proving that large-$n$ extremal cographs arise by pumping a fixed component inside a small core graph. It introduces biclique-profiles and a dynamic-programming approach to enumerate extremal cographs, and establishes a Lifting Theorem that decomposes extremal structures into cores with pumped components. The central Pumping Theorem shows a periodic, linear growth with coefficient $\alpha = s-1 + \frac{t-1}{2}$, and characterizes the regularity and common-neighborhood structure of pumping components. The authors also completely classify $K_{3,3}$-free extremal cographs and provide a DP-based method for enumerating extremal examples for small $n$, highlighting a rigid, star-like extremal geometry induced by the P4-free constraint and the forbidden bicliques.”

Abstract

A cograph is a graph that contains no induced path $P_4$ on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Turán problem restricted to cographs: for fixed integers $s \leq t$, what is the maximum number of edges in an $n$-vertex cograph that does not contain $K_{s,t}$ as a subgraph? This problem falls within the framework of induced Turán numbers $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ introduced by Loh, Tait, Timmons, and Zhou. Our main result is a Pumping Theorem: for every $s\le t$ there exists a period $R$ and core cographs such that for all sufficiently large $n$ an extremal cograph is obtained by repeatedly pumping one designated pumping component inside the appropriate core (depending on $n\bmod R$). We determine the linear coefficient of $\text{ex}(n, \{K_{s,t}, P_4\text{-ind}\})$ to be $s-1 + \frac{t-1}{2}$. Moreover, the pumping components are $(t-1)$-regular and have $s-1$ common neighbours in the respecitve core graphs, giving the extremal cographs a particularly rigid extremal star-like shape. Motivated by the rarity of complete classification of extremal configurations, we completely classify all $K_{3,3}$-free extremal cographs by proof. We also develop a dynamic programming algorithm for enumerating extremal cographs for small $n$.

Bipartite Turán problem on cographs

TL;DR

This work analyzes ex within the restricted framework of cographs, proving that large- extremal cographs arise by pumping a fixed component inside a small core graph. It introduces biclique-profiles and a dynamic-programming approach to enumerate extremal cographs, and establishes a Lifting Theorem that decomposes extremal structures into cores with pumped components. The central Pumping Theorem shows a periodic, linear growth with coefficient , and characterizes the regularity and common-neighborhood structure of pumping components. The authors also completely classify -free extremal cographs and provide a DP-based method for enumerating extremal examples for small , highlighting a rigid, star-like extremal geometry induced by the P4-free constraint and the forbidden bicliques.”

Abstract

A cograph is a graph that contains no induced path on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Turán problem restricted to cographs: for fixed integers , what is the maximum number of edges in an -vertex cograph that does not contain as a subgraph? This problem falls within the framework of induced Turán numbers introduced by Loh, Tait, Timmons, and Zhou. Our main result is a Pumping Theorem: for every there exists a period and core cographs such that for all sufficiently large an extremal cograph is obtained by repeatedly pumping one designated pumping component inside the appropriate core (depending on ). We determine the linear coefficient of to be . Moreover, the pumping components are -regular and have common neighbours in the respecitve core graphs, giving the extremal cographs a particularly rigid extremal star-like shape. Motivated by the rarity of complete classification of extremal configurations, we completely classify all -free extremal cographs by proof. We also develop a dynamic programming algorithm for enumerating extremal cographs for small .
Paper Structure (41 sections, 11 theorems, 48 equations, 1 figure)

This paper contains 41 sections, 11 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Cographs
  4. Main results
  5. Further background and related work
  6. More insights about the Zarankievicz problem.
  7. Structural constraints and improved bounds.
  8. Induced Turán numbers and hereditary properties.
  9. VC-dimension, cographs, and the Erdős-Hajnal conjecture.
  10. Biclique-Profiles
  11. Dynamic programming
  12. The Lifting Theorem
  13. Proofs of the Main Results
  14. Special cases
  15. case 1: $n$ is even.
  16. ...and 26 more sections

Key Result

Theorem 1

Let $s, \ t \in \mathbb{N}$ with $2 \leq s \leq t$. There exist $N_{s,t}, R \in \mathbb{N}$ and core graphs $G_0, \ldots, G_{R-1}$ each of size at most $N_{s,t}$ with designated $(t-1)$-regular components $H_j \subseteq G_j$ with $s-1$ common neighbors outside of $H_j$ such that for any $n$ large en Moreover, there exists at least one residue class $g \in \mathbb{Z} / R \mathbb{Z}$ such that the $

Figures (1)

  • Figure 1: Cotree of a $K_{5,5}$-free extremal cograph on 42 vertices. The root is marked as purple. The corresponding cograph is the product of $K_{2,2}$ together with the sum of four $5$-cliques and three complete multipartite $K_{2,2,2}$.

Theorems & Definitions (26)

  • Definition 1: Pumping
  • Theorem 1: Pumping Theorem
  • Theorem 2: Forbidden stars
  • Theorem 3
  • Theorem 4
  • Definition 2: Biclique-profile
  • Lemma 1: Profile operations
  • proof : Proof of Lemma \ref{['thm:profile_operations']}
  • Lemma 2: Profile restriction
  • proof : Proof of Lemma \ref{['thm:profile_restriction']}
  • ...and 16 more