On the Book Thickness of k-Trees

Every k-tree has book thickness at most k + 1, and this bound is best possible for all k ≥ 3. Vandenbussche et al. [SIAM J. Discrete Math., 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 ≥ 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.