Decompositions of permutations and book embeddings

In the influential paper in which he proved that every graph with m edges can be embedded in a book with O(m1/2) pages, Malitz proved the existence of d-regular n-vertex graphs that require Ω(n 1 2 − 1 d ) pages. In view of the O(m1/2) bound, this last bound is tight when d > log n, and Malitz asked if it is also tight when d < log n. We answer negatively to this question, by showing that there exist d-regular graphs that require Ω(n 1 2 − 1 2(d−1) ) pages. In addition, we show that the bound O(m1/2) is not tight either for most d-regular graphs, by proving that for each fixed d, w.h.p. the random d-regular graph can be embedded in o(m1/2) pages. We also give a simpler proof of Malitz’s O(m1/2) bound, and improve the proportionality constant. As we investigated these questions on book embeddings, we stumbled upon, and shifted our attention to, questions about decompositions of permutations which seem to be of independent interest. For instance, we proved that if A is a k×n-matrix each of whose rows is a random permutation of [n], then w.h.p. there is a column permutation such that in the resulting matrix each row can be decomposed into o(n1/2) monotone decreasing subsequences.