# Maximum Genus of Strong Embeddings

Maximum Genus of Strong Embeddings The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# Maximum Genus of Strong Embeddings

, Volume 19 (3) – Mar 3, 2017

## Maximum Genus of Strong Embeddings

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 437–446 1 2 3 Er-ling Wei ,Yan-pei Liu ,Han Ren Department of Mathematics, Renming University of China, Beijing 100872, China Department of Mathematics, Northern Jiaotong University, Beijing 100044, China Department of mathematics, East China Normal University, Shanghai 200062, China Abstract The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph. Keywords CDC, Halin graph, strong embedding, genus, surface 2000 MR Subject Classiﬁcation 05C10 1 Introduction Terminologies not explained here can always be seen in [8] and [3]. For any graph G,let V, E denote the set of vertices, edges and (G),δ(G) denote the maximum, minimum valence of the graph G respectively. Write E as the set of edges incident with v ∈ V .Let S be a closed surface. An embedding Ψof G into Σ, a polyhedron on S, is a 1-1 mapping Ψ : G → Σ, where each connected component of Σ\G is homomorphic to an open disc, called face of Ψ. The set of faces is denoted by F and write F as the set of faces incident with v ∈ V .And |X| is the cardinality of a set X. A circuit C of G is a connected subgraph with each vertex of valence two. A circuit double cover (CDC) of a graph G is a list of circuits C of G such that each edge of G lies in exactly two members of...

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Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0119-x
Publisher site
See Article on Publisher Site

### Abstract

The strong embedding conjecture states that any 2-connected graph has a strong embedding on some surface. It implies the circuit double cover conjecture: Any 2-connected graph has a circuit double cover. Conversely, it is not true. But for a 3-regular graph, the two conjectures are equivalent. In this paper, a characterization of graphs having a strong embedding with exactly 3 faces, which is the strong embedding of maximum genus, is given. In addition, some graphs with the property are provided. More generally, an upper bound of the maximum genus of strong embeddings of a graph is presented too. Lastly, it is shown that the interpolation theorem is true to planar Halin graph.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

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