$(n,k)$ -可扩图的曲面嵌入
Surface embedding of $(n,k)$-extendable graphs
摘要：
研究匹配可扩图的曲面嵌入问题。完美匹配理论有两个扩展方向，即匹配可拓性和因素关键性。在解决由Plummer提出的问题时，Dean（The matching extensibility of surfaces，J. Combin. Theory Ser.B54（1992），133- 141）建立了一个令人着迷的最小数$k= μ(Σ) $的公式，使得每个$Σ$ -可嵌入图都不是$k$ -可扩的。Su和Zhang，Plummer和Zha找到了使每个$Σ$ -可嵌入图都不是$n$ -因子临界图的最小数目$n=ρ(Σ)$.基于将这两个参数联系起来的$(n,k)$ -图的概念，我们得到了使每个$Σ$ -可嵌入图都不是$(n,k)$ -图的最小数$k=μ(n,Σ)$的公式.为了研究这个双参数问题，我们考虑了它的对偶问题，并反过来求出了$μ(n,Σ)$。同样的方法也适用于重新发现数字$ρ(Σ)$的公式。
英文摘要：
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 (The matching extendability of surfaces, J. Combin. Theory Ser. B 54 (1992), 133--141) established the fascinating formula for the minimum number $k= μ(Σ) $ such that every $Σ$-embeddable graph is not $k$-extendable. Su and Zhang, Plummer and Zha found the minimum number $n=ρ(Σ)$ such that every $Σ$-embeddable graph is not $n$-factor-critical. Based on the notion of $(n,k)$-graphs which associates these two parameters, we found the formula for the minimum number $k=μ(n,Σ)$ such that every $Σ$-embeddable graph is not an $(n,k)$-graph. To access this two-parameter-problem, we consider its dual problem and find out $μ(n,Σ)$ conversely. The same approach works for rediscovering the formula of the number $ρ(Σ)$.
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