Relations To Other Graph Properties
The book thickness of a graph is at most 1 if and only if it is outerplanar. The book thickness is at most two if and only if the graph is a subgraph of a planar graph with a Hamiltonian cycle; for instance, the Goldner–Harary graph, a non-Hamiltonian maximal planar graph, provides an example of a planar graph that does not have book thickness two. Since finding Hamiltonian cycles in maximal planar graphs is NP-complete, so is the problem of testing whether the book thickness is two. All planar graphs have book thickness at most four. It was claimed in that some planar graphs have book thickness exactly four. However, a detailed proof of this claim, announced in a subsequent journal paper, has never been published. Graphs of treewidth k have book thickness at most k + 1. Graphs of genus g have book thickness O(√g).
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