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# Enumerating unlabelled embeddings of digraphs

Published:2017-09-29

• Yichao Chen,
Department of Mathematics, Hunan University, 410082 Changsha, China
• Xiaojian Gao,
Department of Mathematics, Hunan University, 410082 Changsha, China
• Yuanqiu Huang,
Department of Mathematics, Normal University of Hunan, 410081 Changsha, China
 Format: LaTeX MathJax PDF

## Abstract

A $2$-cell embedding of an Eulerian digraph $D$ into a closed surface is said to be directed if the boundary of each face is a directed closed walk in $D$. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles $B_n$, directed dipoles $OD_{2n}$ and for a class of regular tournaments $T_{2n+1}$.
 Keywords: Eulerian digraph, directed embedding, unlabelled embedding
 MSC Classifications: 05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

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