Summary: In the influential papers in which \textit{M. Malitz} [J. Algorithms 17, No. 1, 71--84 (1994; Zbl 0810.68102); ibid. No. 1, 85--109 (1994; Zbl 0810.68103)] proved that every graph with \(m\) edges can be embedded in a book with \(O({m}^{1/2})\) pages, he proved the existence of \(d\)-regular \(n\)-vertex graphs that require \(\Omega(\sqrt{d}n^{\frac{1}{2}-\frac{1}{d}})\) pages. In view of the \(O({m}^{1/2})\) bound, this last bound is tight when \(d > \log{n}\), and Malitz [loc. cit.] 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 \(\Omega(n^{\frac{1}{2}-\frac{1}{2(d-1)}})\) pages. In addition, we show that the bound \(O({m}^{1/2})\) is not tight either for most \(d\)-regular graphs by proving that for each fixed \(d\), with high probability the random \(d\)-regular graph can be embedded in \(o({m}^{1/2})\) pages. We also give a simpler proof of Malitz's \(O({m}^{1/2})\) bound and improve the proportionality constant.