Surface embedding of (n,k)(n,k)-extendable graphs

This paper is concerned with the surface embedding of matching extendable graphs. There are two directions extending the theory of perfect matchings, that is, matching extendability and factor-criticality. In solving a problem posed by Plummer, Dean (1992) established the fascinating formula for the minimum number k=μ(Σ)k=μ(Σ) such that everyΣΣ-embeddable graph is not kk-extendable. Su and Zhang, Plummer and Zha found the minimum number n=ρ(Σ)n=ρ(Σ) such that every ΣΣ-embeddable graph is not nn-factor-critical. Based on the notion of (n,k)(n,k)-graphs which associates these two parameters, we found the formula for the minimum number k=μ(n,Σ)k=μ(n,Σ) such that every ΣΣ-embeddable graph is not an (n,k)(n,k)-graph. To access this two-parameter-problem, we consider its dual problem and find out μ(n,Σ)μ(n,Σ) conversely. The same approach works for rediscovering the formula of the number  ρ(Σ)ρ(Σ).