Star arboricity relaxed book thickness of $K_n$

TL;DR

This paper explores the star arboricity relaxed book thickness of complete graphs, demonstrating that all page-subgraphs can be star forests with minimal additional pages or cross-caps.

Contribution

It introduces a method to ensure all page-subgraphs are star forests in book embeddings of complete graphs with minimal modifications.

Findings

All page-subgraphs can be star forests with one extra cross-cap page.

Alternatively, they can be star forests with two additional ordinary pages.

The approach reduces the complexity of book embeddings for complete graphs.

Abstract

A book embedding of the complete graph $K_n$ needs $\lceil \frac{n}{2} \rceil$ pages and the page-subgraphs can be chosen to be spanning paths (for $n$ even) and one spanning star for $n$ odd. We show that all page-subgraphs can be chosen to be {\rm star forests} by including one extra {\rm cross-cap} page or two new ordinary pages.