On Dispersable Book Embeddings
Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, Michael, Kaufmann, and Sergey Pupyrev

TL;DR
This paper investigates dispersable book embeddings of graphs, disproves a longstanding conjecture for certain regular bipartite graphs, and establishes conditions under which these graphs are dispersable.
Contribution
It disproves Bernhart and Kainen's conjecture for k=3 and k=4 bipartite graphs, providing specific counterexamples and positive results for certain classes.
Findings
Gray graph has dispersable book thickness four
Folkman graph has dispersable book thickness five
3-connected 3-regular bipartite planar graphs are dispersable
Abstract
In a dispersable book embedding, the vertices of a given graph must be ordered along a line l, called spine, and the edges of G must be drawn at different half-planes bounded by l, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding of is referred to as the dispersable book thickness of . Graph is called dispersable if holds (note that always holds). Back in 1979, Bernhart and Kainen conjectured that any -regular bipartite graph is dispersable, i.e., . In this paper, we disprove this conjecture for the cases (with a computer-aided proof), and (with a purely combinatorial proof). In particular, we show that the Gray graph, which…
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