The crossing number of the cone of a graph

TL;DR

This paper investigates the relationship between the crossing number of a graph and its cone, aiming to determine the minimal increase in crossing number when adding a universal vertex, with exact results for small cases.

Contribution

It introduces the function f(k) representing the minimal crossing number of the cone given the original graph's crossing number, providing exact values for simple graphs and asymptotic bounds for multigraphs.

Findings

Exact values of f(k) for small k in simple graphs.

Asymptotic bound f(k)=k+Θ(√k) for multigraphs.

The difference in crossing numbers can be arbitrarily large for fixed crossing number.

Abstract

Motivated by a problem asked by Richter and by the long standing Harary-Hill conjecture, we study the relation between the crossing number of a graph $G$ and the crossing number of its cone $CG$, the graph obtained from $G$ by adding a new vertex adjacent to all the vertices in $G$. Simple examples show that the difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In this work, we are interested in finding the smallest possible difference, that is, for each non-negative integer $k$, find the smallest $f(k)$ for which there exists a graph with crossing number at least $k$ and cone with crossing number $f(k)$. For small values of $k$, we give exact values of $f(k)$ when the problem is restricted to simple graphs, and show that $f(k)=k+\Theta (\sqrt {k})$ when multiple edges are allowed.