Graphs and Combinatorics

, Volume 35, Issue 4, pp 941–957

# The (n, k)-Extendable Graphs in Surfaces

• Yushan Ma
• Qiuli Li
• Heping Zhang
Original Paper

## 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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## Authors and Affiliations

• Yushan Ma
• 1
• Qiuli Li
• 1
• Heping Zhang
• 1
1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China