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Enumerating unlabelled embeddings of digraphs

  • 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
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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 Eulerian digraph, directed embedding, unlabelled embedding
MSC Classifications: 05C10 show english descriptions Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 05C10 - Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
 

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