<p>Let <span class="math-container">$G$</span> be a finite, simple, undirected graph.<br />
<strong>What conditions on <span class="math-container">$G$</span> ensure that every rotation system of <span class="math-container">$G$</span> corresponds to a cellular embedding of <span class="math-container">$G$</span> on an oriented surface of small genus?</strong>
(e.g. genus at most 2)</p>
<p>In particular, <strong>what conditions on a planar graph <span class="math-container">$G$</span> ensure that every rotation system of <span class="math-container">$G$</span> corresponds to a plane embedding of <span class="math-container">$G$</span>?</strong></p>
<p><em>Context:</em> I am interested in rotation systems with a specific property (biparite dual), let us call it property <span class="math-container">$P$</span>.<br />
I would like to find at least a small class of planar graphs <span class="math-container">$G$</span> such that it suffices to check all plane embeddings of <span class="math-container">$G$</span> in order to test existence of rotation systems of <span class="math-container">$G$</span> with property <span class="math-container">$P$</span>.</p>

CC BY-SA 4.0
2023-06-09T06:45:25.847Z
The keyword you're looking for is "Maximum genus embedding", and this page might also be helpful : https://en.wikipedia.org/wiki/Xuong_tree
Louis Esperet
CC BY-SA 4.0
2023-06-09T07:19:08.520Z
@LouisEsperet, Thank you for your comment.
Cyriac Antony
CC BY-SA 4.0
2023-06-09T07:28:34.523Z
The notion of maximum genus indeed answers the question. This motivates another question. For a constant g (such as g=2), what conditions on $G$ ensure that every rotation system of $G$ corresponds to an *almost cellular embedding* of $G$ on an oriented surface of genus at most g? (by almost cellular embedding, I mean an embedding in which every face except possibly one is homemorphic to a disk).
Cyriac Antony
<graph-theory><planar-graphs><embeddings><topological-graph-theory>
Graphs such that every rotation system admits an embedding on a surface of small genus
Cyriac Antony
# Graphs such that every rotation system admits an embedding on a surface of small genus

(Asked by: username_0 on 2023-06-09)

Let $G$ be a finite, simple, undirected graph.  
**What conditions on $G$ ensure that every rotation system of $G$ corresponds to a cellular embedding of $G$ on an oriented surface of small genus?**
(e.g. genus at most 2)

In particular, **what conditions on a planar graph $G$ ensure that every rotation system of $G$ corresponds to a plane embedding of $G$?**

*Context:* I am interested in rotation systems with a specific property (biparite dual), let us call it property $P$.  
I would like to find at least a small class of planar graphs $G$ such that it suffices to check all plane embeddings of $G$ in order to test existence of rotation systems of $G$ with property $P$.

**Tags:** graph-theory, planar-graphs, embeddings, topological-graph-theory
> **Comments:**
> **username_1**: The keyword you're looking for is "Maximum genus embedding", and this page might also be helpful : https://en.wikipedia.org/wiki/Xuong_tree  
> **username_0**: @LouisEsperet, Thank you for your comment.  
> **username_0**: The notion of maximum genus indeed answers the question. This motivates another question. For a constant g (such as g=2), what conditions on $G$ ensure that every rotation system of $G$ corresponds to an *almost cellular embedding* of $G$ on an oriented surface of genus at most g? (by almost cellular embedding, I mean an embedding in which every face except possibly one is homemorphic to a disk).  

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## 0 answers

