All bipartite circulants are dispersable
We show that a cyclic vertex order due to Yu, Shao and Li gives a dispersable book embedding for any bipartite circulant.
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arXiv subjects
Shannon Overbay
Publications and source records attributed to Shannon Overbay.
On dispersability of some circulant graphs
The matching book thickness of a graph is the least number of pages in a book embedding such that each page is a matching. A graph is dispersable if its matching book thickness equals its maximum degree. Minimum page matching book embeddings are given for bipartite and for most non-bipartite circulants contained in the (Harary) cube of a cycle and for various higher-powers.
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Graph embeddings with no Hamiltonian extensions
We show that extending an embedding of a graph $Γ$ in a surface to an embedding of a Hamiltonian supergraph can be blocked by certain planar subgraphs but, for some subdivisions of $Γ$, Hamiltonian extensions must exist.
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Book embeddings of graphs and a theorem of Whitney
It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page book embedding. The latter extends a theorem of H. Whitney and gives two-page book embeddings for $X$-trees and square grids.
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On dispersability of some products of cycles
We show that the matching book thickness of the Cartesian product of two odd-length cycle-graphs is five if at least one of the cycles has length 3 or 5.
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Cubic planar bipartite graphs are dispersable
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.
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