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Graphs and Combinatorics

, Volume 35, Issue 4, pp 941–957 | Cite as

The (nk)-Extendable Graphs in Surfaces

  • Yushan Ma
  • Qiuli Li
  • Heping ZhangEmail author
Original Paper
  • 28 Downloads

Abstract

A connected graph G with at least \(2k+2\) vertices is called k-extendable if it has a matching of size k and every such matching can extend to a perfect matching in G. A graph G is an (nk)-graph if for any set S of n vertices, the subgraph \(G-S\) is k-extendable. For a surface \(\Sigma \), Lu and Wang established the pleasurable formula of the minimum number \(k=\mu (n,{\varSigma })\) such that no \({\varSigma }\)-embeddable graph is an (nk)-graph by fixing the parameter n. In this paper, we make the other parameter k fixed and figure out the minimum number \(n=\rho (k,{\varSigma })\) such that no \({\varSigma }\)-embeddable graph is an (nk)-graph. By the way, the well-known Dean’s formula (Theorem 1.1) can be reproduced. Further, we find out an upper bound on the order of (nk)-graphs embedded in \({\varSigma }\) for any non-negative integers nk with \(k\ge 1\), \(n+2k\ge 6\) and \((n,k)\ne (0,3)\), which yields two results: (i) there are infinitely many (nk)-graphs embedded in the sphere (resp. the projective plane) if and only if \(n+2k\le 4\); (ii) for each surface \({\varSigma }\) homeomorphic to neither the sphere nor the projective plane, there are infinitely many \({\varSigma }\)-embeddable (nk)-graphs if and only if \(n+2k\le 5\) or \((n,k)=(0,3)\).

Keywords

Perfect matching Matching extendability Factor-criticality \((n{, }k)\)-graph Graph in surface 

Notes

Acknowledgements

The authors sincerely thank three referees for their careful reading and valuable comments and suggestions.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

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