Improvement on the decay of crossing numbers

TL;DR

This paper demonstrates that for large graphs with many edges, it is possible to find subgraphs that retain most of the original crossing number while having fewer edges, showing a continuous decay property.

Contribution

It generalizes previous results by proving a continuous decay of crossing numbers in dense graphs with many edges.

Findings

Subgraphs can preserve most of the crossing number despite having fewer edges.

The decay of crossing numbers is continuous for graphs with sufficiently many edges.

The result extends prior work by Fox and Toth on crossing number decay.

Abstract

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a subgraph G' of at most (1-delta)m edges and crossing number at least (1-epsilon)cr(G). This generalizes the result of J. Fox and Cs. Toth.