We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl's result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels.
Bibliographical noteFunding Information:
Yichao Chen is supported by the NNSFC under Grant No. 11471106 . Jonathan Gross is supported by Simons Foundation Grant No. 315001 .
© 2017 Elsevier B.V.
- Asymptotic values
- Genus distribution
- Symmetric group
- Wheel graph
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics