# An Euler‐genus approach to the calculation of the crosscap‐number polynomial

An Euler‐genus approach to the calculation of the crosscap‐number polynomial In 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Graph Theory Wiley

# An Euler‐genus approach to the calculation of the crosscap‐number polynomial

, Volume 88 (1) – Jan 1, 2018
21 pages

/lp/wiley/an-euler-genus-approach-to-the-calculation-of-the-crosscap-number-Y2JtnMHbDc
Publisher
Wiley
ISSN
0364-9024
eISSN
1097-0118
D.O.I.
10.1002/jgt.22186
Publisher site
See Article on Publisher Site

### Abstract

In 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial.

### Journal

Journal of Graph TheoryWiley

Published: Jan 1, 2018

Keywords: ; ; ; ; ; ;

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