Title  The graphs with a symmetrical Euler cycle 
Author  
Corresponding Author  Praeger，Cheryl E. 
Publication Years  2022

DOI  
Source Title  
EISSN  25909770

Volume  5Issue:3 
Abstract  The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e, e,..., e) of these ℓ edges, each occurring exactly once, such that e, e are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle is symmetrical if there exists an automorphism of the graph such that e → e for each i. The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or biregular on edges, respectively. Symmetrical Euler cycles arise naturally from arctransitive embeddings of graphs in surfaces since, for each face of the embedded graph, the sequence of edges on the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this edgeset. We first classify all finite connected graphs which admit a cyclic subgroup of automorphisms that is regular or biregular on edges, and identify more than a dozen infinite families of examples. We then prove that exactly six of these families consist of graphs with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arctransitive maps. 
Keywords  
URL  [Source Record] 
Language  English

SUSTech Authorship  Others

Funding Project  National Natural Science Foundation of China[11771200];National Natural Science Foundation of China[11931005];National Natural Science Foundation of China[12101518];National Natural Science Foundation of China[61771019];Australian Research Council[DP160102323];Society for French Studies[ZR2020MA044];

Scopus EID  2s2.085134495499

Data Source  Scopus

Citation statistics 
Cited Times [WOS]:0

Document Type  Journal Article 
Identifier  http://kc.sustech.edu.cn/handle/2SGJ60CL/359590 
Department  Department of Mathematics 深圳国际数学中心（杰曼诺夫数学中心）（筹） 
Affiliation  1.School of Mathematical Sciences,Xiamen University,Xiamen,China 2.Department of Mathematics,SUSTech International Center for Mathematics,Southern University of Science and Technology,Shenzhen,China 3.Department of Mathematics and Statistics,The University of Western Australia,Crawley,6009,Australia 4.School of Mathematics and Information Science,Yantai University,Yantai,China 
Recommended Citation GB/T 7714 
Chen，Jiyong,Li，Cai Heng,Praeger，Cheryl E.,et al. The graphs with a symmetrical Euler cycle[J]. Art of Discrete and Applied Mathematics,2022,5(3).

APA 
Chen，Jiyong,Li，Cai Heng,Praeger，Cheryl E.,&Song，Shu Jiao.(2022).The graphs with a symmetrical Euler cycle.Art of Discrete and Applied Mathematics,5(3).

MLA 
Chen，Jiyong,et al."The graphs with a symmetrical Euler cycle".Art of Discrete and Applied Mathematics 5.3(2022).

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