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Graphs and Combinatorics

, Volume 34, Issue 5, pp 1089–1099 | Cite as

The Matching Extendability of Optimal 1-Planar Graphs

  • Jun Fujisawa
  • Keita Segawa
  • Yusuke Suzuki
Original Paper
  • 64 Downloads

Abstract

A graph G is said to be 1-planar if it can be drawn on the sphere or plane so that any edge of G has at most one crossing point with another edge. Moreover, G is called an optimal 1-planar graph if \(|E(G)| = 4|V(G)|-8\). In this paper, we investigate the matching extendability of optimal 1-planar graphs. It is shown that every optimal 1-planar graph G of even order is 2-extendable unless G contains a 4-cycle C which separates the graph into two odd components. Moreover, for any 5-connected optimal 1-planar graph, we characterize a matching with three edges which is not extendable.

Keywords

Optimal 1-planar graph Perfect matching Extendability 

Notes

Acknowledgements

The authors would like to thank Katsuhiro Ota whose comment led to significant improvement in the proof of Lemma 4.

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Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Business and CommerceKeio UniversityYokohamaJapan
  2. 2.Graduate School of Science and TechnologyNiigata UniversityNiigataJapan
  3. 3.Department of MathematicsNiigata UniversityNiigataJapan

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