On the genus distributions of wheels and of related graphs

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 su...

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Bibliographic Details
Published inDiscrete mathematics Vol. 341; no. 4; pp. 934 - 945
Main Authors Chen, Yichao, Gross, Jonathan L., Mansour, Toufik
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.04.2018
Subjects
Asymptotic values
Genus distribution
Symmetric group
Wheel graph
Asymptotic values
Wheel graph
Symmetric group
Genus distribution
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Summary: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.
ISSN:0012-365X
1872-681X
DOI:10.1016/j.disc.2017.12.007