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


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.


Optimal 1-planar graph Perfect matching Extendability 



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


  1. 1.
    Aldred, R.E.L., Kawarabayashi, K., Plummer, M.D.: On the matching extendability of graphs in surfaces. J. Comb. Theory Ser. B 98, 105–115 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldred, R.E.L., Plummer, M.D.: Edge proximity and matching extension in planar triangulations. Australas. J. Comb. 29, 215–224 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aldred, R.E.L., Plummer, M.D.: Proximity thresholds for matching extension in planar and projective planar triangulations. J. Graph Theory 67, 38–46 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auer, C., Bachmaier, C., Brandenburg, F.J., Gleißner, A., Hanauer, K., Neuwirth, D., Reislhuber, J.: Outer 1-planar graphs. Algorithmica 74, 1293–1320 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, Z.-Z., Kouno, M.: A linear-time algorithm for 7-coloring 1-plane graphs. Algorithmica 43, 147–177 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Czap, J., Hudák, D.: On drawings and decompositions of 1-planar graphs. Electron. J. Combin. 20(2), 54 (2013)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)Google Scholar
  8. 8.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Math. 307, 854–865 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kawarabayashi, K., Negami, S., Plummer, M.D., Suzuki, Y.: The 2-extendability of 5-connected graphs on surfaces with large representativity. J. Comb. Theory Ser. B 101, 206–213 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. J. Graph Theory 72, 30–71 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Noguchi, K., Suzuki, Y.: Relationship among triangulations, quadrangulations and optimal 1-planar graphs. Graphs Comb. 31, 1965–1972 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Plummer, M.D.: A theorem on matchings in the plane. Ann. Discrete Math. 41, 347–354 (1989)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Plummer, M.D.: Extending matchings in planar graphs IV. Discrete Math. 109, 207–219 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Plummer, M.D.: Recent Progress in Matching Extension, Building Bridges. Bolyai Society Mathematical Studies, vol. 19, pp. 427–454. Springer, Berlin (2008)Google Scholar
  15. 15.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Semin. Univ. Hamburg 29, 107–117 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Suzuki, Y.: Re-embeddings of maximum 1-planar graphs. SIAM J. Discrete Math. 24, 1527–1540 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Thomas, R., Yu, X.: 4-connected projective-planar graphs are Hamiltonian. J. Comb. Theory Ser. B 62, 114–132 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations