A polynomial resultant approach to algebraic constructions of extremal graphs

TL;DR

This paper introduces a polynomial resultant method from algebraic geometry to construct explicit extremal graphs, advancing the understanding of Turán numbers for bipartite graphs and related hypergraphs.

Contribution

It develops a novel algebraic approach using multipolynomial resultants for explicit extremal graph construction, improving bounds and addressing open problems in extremal graph theory.

Findings

Matched lower bounds for Turán number of 1-subdivision of K_{3,t_1}

Linear Turán number of Berge theta hypergraph with t_2=217

Improved bounds on minimal t_1 for extremal constructions

Abstract

The Tur\'{a}n 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 Tur\'{a}n number of $1$-subdivision of $K_{3,t_{1}}$ and linear Tur\'{a}n number of Berge theta hyerpgraph $\Theta_{3,t_{2}}^{B}$ 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].