Cubic planar bipartite graphs are dispersable

TL;DR

This paper proves that all cubic planar bipartite multigraphs are dispersable, confirming a conjecture for this class and expanding understanding of book embeddings in graph theory.

Contribution

It establishes that every cubic planar bipartite multigraph is dispersable, relaxing previous connectivity conditions and confirming a key conjecture in graph embedding theory.

Findings

All cubic planar bipartite multigraphs are dispersable.

Disproves the conjecture that only 3-connected graphs are dispersable.

Provides new references and context in the postscript.

Abstract

A graph is called dispersable if it has a book embedding in which each page has maximum degree 1 and the number of pages is the maximum degree. Bernhart and Kainen conjectured every k-regular bipartite graph is dispersable. Forty years later, Alam, Bekos, Gronemann, Kaufmann, and Pupyrev have disproved this conjecture, identifying nonplanar 3- and 4-regular bipartite graphs that are not dispersable. They also proved all cubic planar bipartite 3-connected graphs are dispersable and conjectured that the connectivity condition could be relaxed. We prove that every cubic planar bipartite multigraph is dispersable. A postscript is added which includes new references.