On the book thickness of $k$-trees

TL;DR

This paper investigates the book thickness of $k$-trees, establishing the tight bounds for certain classes of $k$-trees with specific tree decompositions, and solving an open problem from prior research.

Contribution

It proves the optimality of previous bounds for $k$-trees with smooth degree-4 tree decompositions, demonstrating the existence of $k$-trees with higher book thickness.

Findings

Constructed $k$-trees with book thickness $k+1$ and smooth degree-4 tree decompositions

Confirmed the bounds are tight for $k eq 3$

Solved an open problem from Vandenbussche et al. (2009)

Abstract

Every $k$-tree has book thickness at most $k+1$, and this bound is best possible for all $k\geq3$. Vandenbussche et al. (2009) proved that every $k$-tree that has a smooth degree-3 tree decomposition with width $k$ has book thickness at most $k$. We prove this result is best possible for $k\geq 4$, by constructing a $k$-tree with book thickness $k+1$ that has a smooth degree-4 tree decomposition with width $k$. This solves an open problem of Vandenbussche et al. (2009)