On dispersable book embeddings

Abstract

In a dispersable book embedding, the vertices of a given graph G must be ordered along a line ℓ, called spine, and the edges of G must be drawn in 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 graph induced by the edges of each page is 1-regular (or equivalently, a matching). The minimum number of pages needed by any dispersable book embedding of G is referred to as the dispersable book thicknessdbt(G) of G. Graph G is called dispersable if dbt(G)=Δ(G) holds (note that Δ(G)≤dbt(G) always holds).Back in 1979, Bernhart and Kainen conjectured that any Δ-regular bipartite graph G is dispersable, i.e., dbt(G)=Δ. In this paper, we employ a counting argument to disprove this conjecture for any fixed value of Δ≥3. Additionally, for the cases Δ=3 and Δ=4 we present concrete counterexamples to the conjecture. In particular, we show that the Gray graph, which is 3-regular and bipartite, has dispersable book thickness four (with a computer-aided proof), while the Folkman graph, which is 4-regular and bipartite, has dispersable book thickness five (with a purely combinatorial proof). On the positive side, we prove that 3-regular bipartite planar graphs are dispersable.