Genus polynomials and crosscap‐number polynomials for ring‐like graphs

Yichao Chen and Jonathan L. Gross

Mathematische Nachrichten, 2019, vol. 292, issue 4, 760-776

Abstract: An H‐linear graph is obtained by transforming a collection of copies of a fixed graph H into a chain. An H‐ring‐like graph is formed by binding the two end‐copies of H in such a chain to each other. Genus polynomials have been calculated for bindings of several kinds. In this paper, we substantially generalize the rules for constructing sequences of H‐ring‐like graphs from sequences of H‐linear graphs, and we give a general method for obtaining a recursion for the genus polynomials of the graphs in a sequence of ring‐like graphs. We use Chebyshev polynomials to obtain explicit formulas for the genus polynomials of several such sequences. We also give methods for obtaining recursions for partial genus polynomials and for crosscap‐number polynomials of a bar‐ring of a sequence of disjoint graphs.

Date: 2019
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https://doi.org/10.1002/mana.201800132

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