A Note on the Crossing Numbers of 5-Regular Graphs
The crossing number cr(G) of a graph G is the smallest number of edge crossings in any drawing of G. In this paper, we prove that there exists a unique 5-regular graph G on 10 vertices with cr(G) = 2. This answers a question by Chia and Gan in the negative. In addition, we also give a new proof of C...
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Sciendo
2020-11-01
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| Series: | Discussiones Mathematicae Graph Theory |
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| Online Access: | https://doi.org/10.7151/dmgt.2203 |
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doaj-65ae397b86bd45b6a47ee776bd79418c2021-09-05T17:20:25ZengSciendoDiscussiones Mathematicae Graph Theory2083-58922020-11-014041127114010.7151/dmgt.2203dmgt.2203A Note on the Crossing Numbers of 5-Regular GraphsOuyang Zhangdong0Department of Mathematics, Hunan First Normal University, Changsha410205, P.R.ChinaThe crossing number cr(G) of a graph G is the smallest number of edge crossings in any drawing of G. In this paper, we prove that there exists a unique 5-regular graph G on 10 vertices with cr(G) = 2. This answers a question by Chia and Gan in the negative. In addition, we also give a new proof of Chia and Gan’s result which states that if G is a non-planar 5-regular graph on 12 vertices, then cr(G) ≥ 2.https://doi.org/10.7151/dmgt.2203crossing number5-regular graphdrawing05c1005c62 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Ouyang Zhangdong |
| spellingShingle |
Ouyang Zhangdong A Note on the Crossing Numbers of 5-Regular Graphs Discussiones Mathematicae Graph Theory crossing number 5-regular graph drawing 05c10 05c62 |
| author_facet |
Ouyang Zhangdong |
| author_sort |
Ouyang Zhangdong |
| title |
A Note on the Crossing Numbers of 5-Regular Graphs |
| title_short |
A Note on the Crossing Numbers of 5-Regular Graphs |
| title_full |
A Note on the Crossing Numbers of 5-Regular Graphs |
| title_fullStr |
A Note on the Crossing Numbers of 5-Regular Graphs |
| title_full_unstemmed |
A Note on the Crossing Numbers of 5-Regular Graphs |
| title_sort |
note on the crossing numbers of 5-regular graphs |
| publisher |
Sciendo |
| series |
Discussiones Mathematicae Graph Theory |
| issn |
2083-5892 |
| publishDate |
2020-11-01 |
| description |
The crossing number cr(G) of a graph G is the smallest number of edge crossings in any drawing of G. In this paper, we prove that there exists a unique 5-regular graph G on 10 vertices with cr(G) = 2. This answers a question by Chia and Gan in the negative. In addition, we also give a new proof of Chia and Gan’s result which states that if G is a non-planar 5-regular graph on 12 vertices, then cr(G) ≥ 2. |
| topic |
crossing number 5-regular graph drawing 05c10 05c62 |
| url |
https://doi.org/10.7151/dmgt.2203 |
| work_keys_str_mv |
AT ouyangzhangdong anoteonthecrossingnumbersof5regulargraphs AT ouyangzhangdong noteonthecrossingnumbersof5regulargraphs |
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1717786358387310592 |