An Euler‐genus approach to the calculation of the crosscap‐number polynomial. Issue 1 (12th September 2017)

• Record Type: Journal Article
• Authors: Chen, Yichao; Gross, Jonathan L.
• Is Part Of: Journal of graph theory. Volume 88:Issue 1(2018)
• Page Start: 80
• Page End: 100
• Publication Date: 2017-09-12
• Subjects: crosscap‐number polynomial -- Euler‐genus distribution -- Euler‐genus polynomial -- genus distribution -- H‐linear families with spiders -- 30B70 -- 42C05; Graph theory -- Periodicals; 511
• DOI: 10.1002/jgt.22186
• Languages: English
• ISSNs: 0364-9024

Abstract

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.