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

TL;DR

This paper investigates the surface embedding properties of $(n,k)$-extendable graphs, deriving formulas for parameters that determine when such graphs are not extendable or factor-critical on various surfaces.

Contribution

It introduces a formula for the minimum $k$ such that $ ext{every } ext{embeddable graph}$ is not $(n,k)$-extendable, extending prior results on matching extendability and factor-criticality.

Findings

Derived the formula for $ ext{minimum }k= ext{mu}(n, ext{Sigma})$ for surface-embeddable graphs.

Revealed the dual relationship between parameters $ ext{mu}(n, ext{Sigma})$ and $ ho( ext{Sigma})$.

Unified the understanding of extendability and factor-criticality on surfaces.

Abstract

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= \mu (\Sigma) $ such that every $\Sigma$-embeddable graph is not $k$-extendable. Su and Zhang, Plummer and Zha found the minimum number $n=\rho(\Sigma)$ such that every $\Sigma$-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 $k=\mu(n,\Sigma)$ such that every $\Sigma$-embeddable graph is not an $(n,k)$-graph. To access this two-parameter-problem, we consider its dual problem and find out $\mu(n,\Sigma)$ conversely. The same approach works for rediscovering the formula of the number $\rho(\Sigma)$.