The Maximum of the Maximum Rectilinear Crossing Numbers of d-regular Graphs of Order n

TL;DR

This paper extends known results on maximum rectilinear crossing numbers to d-regular graphs, introduces star-like drawings that maximize crossings, and conjectures optimality for certain cases, providing simpler proofs for some prior results.

Contribution

It generalizes maximum rectilinear crossing number results to d-regular graphs and introduces star-like drawings that are conjectured to be optimal.

Findings

Star-like drawings maximize rectilinear crossing numbers for certain d-regular graphs.

A simpler proof of existing results by Furry and Kleitman.

Conjecture that star-like drawings are optimal for all cases.

Abstract

We extend known results regarding the maximum rectilinear crossing number of the cycle graph (C_n) and the complete graph (K_n) to the class of general d-regular graphs R_{n,d}. We present the generalized star drawings of the d-regular graphs S_{n,d} of order n where n+d= 1 mod 2 and prove that they maximize the maximum rectilinear crossing numbers. A star-like drawing of S_{n,d} for n = d = 0 mod 2 is introduced and we conjecture that this drawing maximizes the maximum rectilinear crossing numbers, too. We offer a simpler proof of two results initially proved by Furry and Kleitman as partial results in the direction of this conjecture.