Genus distributions of iterated 3-wheels and 3-prisms

Mehvish I. Poshni, Imran F. Khan, Jonathan L. Gross


The iterated 3-prism Pr3n is the cartesian product C3□ Pn of a 3-cycle and an path. At each end of the iterated 3-prism, there is a 3-cycle whose vertices are 3-valent in C3□ Pn. The iterated 3-wheel W3n is obtained by contracting one of these 3-cycles in C3□ Pn + 1 to a single vertex. Using rooted-graphs, we derive simultaneous recursions for the partitioned genus distributions of W3n and a formula for the genus distribution of the graphs Pr3n. A seemingly straightforward way to construct either the sequence of iterated prisms Pr3n or the sequence of iterated wheels W3n, would be by iterative amalgamation of a copy of C3□ K2, such that a copy of C3 contained in it is matched to the “newest” copy of C3 in the growing graph. Calculating genus distributions for the sequences would then involve an excessively large set of simultaneous recurrences. To avoid this, we propose a method of iterative surgery, under which the same vertex is considered a root-vertex in all graphs of the sequence, and in which the successive calculations of genus distributions require only four simultaneous recurrences. We also prove that the genus distribution of Pr3n not only dominates the genus distribution of W3n − 1, but is also dominated by the genus distribution of W3n.


Genus distribution, rooted-graph, production, partitioned genus distribution, 3-prism, 3-wheel.

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ISSN: 1855-3974

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