Unexpected behaviour of crossing sequences

TL;DR

This paper constructs specific graphs demonstrating unexpected crossing number behaviors across different surfaces, supporting a conjecture and resolving a known problem in topological graph theory.

Contribution

It proves the existence of graphs with prescribed crossing numbers on surfaces of varying genus, revealing surprising behaviors in crossing sequences.

Findings

Existence of graphs with cr_0(G)=a, cr_1(G)=b, cr_2(G)=0 for a>b>0

Supports a conjecture of Archdeacon et al.

Resolves a problem posed by Salazar.

Abstract

The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.