Expected Crossing Numbers

TL;DR

This paper investigates the expected weighted crossing number of random graphs with independent uniform edge weights, introducing a variation of the Crossing Lemma tailored for expectations.

Contribution

It presents a new variation of the Crossing Lemma applicable to expected values and analyzes the expected crossing number for graphs with uniformly distributed random edge weights.

Findings

Derived a variation of the Crossing Lemma for expectations.

Analyzed the expected crossing number for graphs with uniform edge weights.

Provided theoretical bounds and properties for the expected crossing number.

Abstract

The expected value for the weighted crossing number of a randomly weighted graph is studied. A variation of the Crossing Lemma for expectations is proved. We focus on the case where the edge-weights are independent random variables that are uniformly distributed on [0,1].