Weighted Turan Problems with Applications

TL;DR

This paper investigates weighted extremal problems in complete graphs, focusing on two types of weights with applications to multipartite extremal problems and the rectilinear crossing number of diameter-4 trees.

Contribution

It introduces the concept of weighted extremal numbers and explores their applications in multipartite graphs and geometric graph theory.

Findings

Derived bounds for weighted extremal numbers in specific graph classes

Applied weighted extremal concepts to rectilinear crossing problems

Connected weighted extremal problems to classical extremal graph theory

Abstract

Suppose the edges of $K_n$ are assigned weights by a weight function $w$. We define the {\em weighted extremal number} \[ \mathrm{ex}(n,w,F):=\max\{w(G)\mid G\subseteq K_n,\text{ and }G\text{ is }F\text{-free}\} \] where $w(G):=\sum_{e\in E(G)}w(e)$. In this paper we study this problem for two types of weights $w$, each of which has an application. The first application is to an extremal problem in a complete multipartite host graph. The second application is to the maximum rectilinear crossing number of trees of diameter 4.