On dispersable book embeddings

In a dispersable book embedding, the vertices of given graph G must be ordered along line ℓ, called spine, and edges drawn in different half-planes bounded by pages book, such that: (i) no two same page cross, (ii) induced each is 1-regular (or equivalently, matching). The minimum number needed any embedding referred to as thickness dbt(G) G. Graph if dbt(G)=Δ(G) holds (note that Δ(G)≤dbt(G) always holds). Back 1979, Bernhart Kainen conjectured Δ-regular bipartite dispersable, i.e., dbt(G)=Δ. this paper, we employ counting argument disprove conjecture for fixed value Δ≥3. Additionally, cases Δ=3 Δ=4 present concrete counterexamples conjecture. particular, show Gray graph, which 3-regular bipartite, has four (with computer-aided proof), while Folkman 4-regular five purely combinatorial proof). On positive side, prove planar graphs are dispersable.