A note on the cross-index of a complete graph based on a linear tree

TL;DR

This paper demonstrates that for any odd number of vertices greater than or equal to 7, a complete graph can be optimally diagrammed with a specific linear tree structure, matching Guy's crossing number formula.

Contribution

It establishes the existence of optimal diagrams for complete graphs with a free maximal linear tree and no free Hamiltonian cycles for odd n ≥ 7.

Findings

Optimal diagram exists for complete graphs with odd n ≥ 7

Crossing number matches Guy's formula in these diagrams

Diagrams can be constructed with a free maximal linear tree

Abstract

In this paper it is shown that a complete graph with $n$ vertices has an optimal diagram, i.e., a diagram whose crossing number equals the value of Guy's formula, with a free maximal linear tree and without free hamiltonian cycles for any odd integer $n \ge 7$.