Received: 13 May 2016 Revised: 9 August 2017 Accepted: 20 August 2017
An Euler-genus approach to the calculation
of the crosscap-number polynomial
Jonathan L. Gross
Department of Mathematics, Hunan
University, 410082 Changsha, China
Department of Computer Science, Columbia
University, New York, NY 10027
Yichao Chen, Department of Mathe-
matics, Hunan University, 410082 Changsha,
China. Email: firstname.lastname@example.org
Contract grant sponsor: NNSFC; contract grant
number: 11471106 (to Y. C.).
Contract grant sponsor: Simons Foundation;
contract grant number: 315001 (to J. G.).
In 1994, J. Chen, J. Gross, and R. Rieper demonstrated
how to use the rank of Mohar's overlap matrix to calcu-
late the crosscap-number distribution, that is, the distribu-
tion of the embeddings of a graph in the nonorientable sur-
faces. 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 nonori-
entable embeddings, without using the overlap matrix. The
crosscap-number polynomial for the nonorientable embed-
dings is then easily calculated from the Euler-genus poly-
nomial and the genus polynomial.
crosscap-number polynomial, Euler-genus distribution, Euler-genus poly-
nomial, genus distribution, -linear families with spiders
1991 MATHEMATICS SUBJECT CLASSIFICA-
TION. PRIMARY: 05C10; SECONDARY:
A surface with Euler characteristic is said to have Euler genus 2−. Thus, the Euler genus is the
crosscap number of any nonorientable surface and twice the genus of any orientable surface. For
the most part, the Euler genus has previously been something to be calculated from the genus and
the crosscap number.
Heretofore, the usual way to calculate the crosscap-number distribution for a graph has been, as
initiated by , by means of the overlap matrix . In this article, we provide a new three-step way
to calculate crosscap-number distribution. The Euler-genus polynomial of a graph is the generating
function for the numbers of embeddings of that graph, according to the Euler genera of surfaces. If
the genus polynomial is known, then from it and the Euler-genus polynomial, we readily obtain the
80 © 2017 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/jgt J Graph Theory. 2018;88:80–100.