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A polynomial resultant approach to algebraic constructions of extremal graphs

Tao Zhang, Zixiang Xu, Gennian Ge

Abstract

The Turán problem asks for the largest number of edges ex$(n,H)$ in an $n$-vertex graph not containing a fixed forbidden subgraph $H$, which is one of the most important problems in extremal graph theory. However the order of magnitude of ex$(n,H)$ for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Turán number of $1$-subdivision of $K_{3,t_{1}}$ and linear Turán number of Berge theta hyerpgraph $Θ_{3,t_{2}}^{B}$ with $t_{1}=25$ and $t_{2}=217$. Moreover, the constant $t_{1}$ improves the random algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of $t_{1}$ concerning a problem posed by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions, Combinatorica, to appear], while the constant $t_{2}$ improves a result of He and Tait~[Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481].

A polynomial resultant approach to algebraic constructions of extremal graphs

Abstract

The Turán problem asks for the largest number of edges ex in an -vertex graph not containing a fixed forbidden subgraph , which is one of the most important problems in extremal graph theory. However the order of magnitude of ex for bipartite graphs is known only in a handful of cases. In particular, giving explicit constructions of extremal graphs is very challenging in this field. In this paper, we develop a polynomail resultant approach to algebraic construction of explicit extremal graphs, which can efficiently decide whether a specified structure exists. A key insight in our approach is the multipolynomial resultant, which is a fundamental tool of computational algebraic geometry. Our main results include the matched lowers bounds for Turán number of -subdivision of and linear Turán number of Berge theta hyerpgraph with and . Moreover, the constant improves the random algebraic construction of Bukh and Conlon~[Rational exponents in extremal graph theory, J. Eur. Math. Soc. 20 (2018), 1747-1757] and makes progress on the known estimation for the smallest value of concerning a problem posed by Conlon, Janzer and Lee ~[More on the extremal number of subdivisions, Combinatorica, to appear], while the constant improves a result of He and Tait~[Hypergraphs with few berge paths of fixed length between vertices, SIAM J. Discrete Math., 33(3), 1472-1481].

Paper Structure

This paper contains 13 sections, 14 theorems, 106 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main technique employed
  3. Constructions of $K_{3,25}'$-free graphs
  4. $a_{1}=b_{1}$ or $a_{1}=c_{1}$ or $b_{1}=c_{1}$
  5. $a_{1}\ne b_{1}$, $a_{1}\ne c_{1}$, $b_{1}\ne c_{1}$ and $a_{2}=b_{2}$ or $a_{2}=c_{2}$ or $b_{2}=c_{2}$
  6. $a_{1}\ne b_{1}$, $a_{1}\ne c_{1}$, $b_{1}\ne c_{1}$, $a_{2}\ne b_{2}$, $a_{2}\ne c_{2}$ and $b_{2}\ne c_{2}$
  7. Proof of Theorem \ref{['subdthm']}
  8. Constructions of $\Theta_{3,217}^{B}$-free linear hypergraphs
  9. $(1,2,1,2)$-type Berge $3$-paths
  10. $(1,2,3,1)$-type Berge $3$-paths
  11. $(1,2,3,2)$-type Berge $3$-paths
  12. Proof of Theorem \ref{['thetathm']}
  13. Concluding remarks

Key Result

Theorem 1.2

$\textup{ex}(n,K_{3,25}')=\Theta(n^{\frac{4}{3}})$.

Theorems & Definitions (36)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Bézout's theorem
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 26 more