Star arboricity relaxed book thickness of $K_n$
Paul C. Kainen
TL;DR
We address decomposing $E(K_n)$ into star forests within a book embedding, seeking the minimum number of pages. Classical results give $bt(K_n)=\lceil \frac{n}{2} \rceil$, realized by Hamiltonian paths for even $n$ or a single spanning star for odd $n$. We introduce a relaxed embedding allowing one cross-cap page, and show $sarbt(K_n)=1+\lceil \frac{n}{2} \rceil$ for $n\ge 4$, with an explicit construction for even $n=2r$ using $r$ cross-cap edges and $r$ two-star pages totaling $|E(K_n)|=2r^2-r$. The paper also discusses related bounds for strict embeddings, connections to the octahedron, and broader implications for sparse pages and related graph families.
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.