On minimal k-factor-critical planar graphs

TL;DR

This paper proves that in planar graphs, minimal k-factor-critical graphs always have a minimum degree of k+1, confirming Favaron and Shi's conjecture for this class of graphs.

Contribution

The paper confirms Favaron and Shi's conjecture for minimal k-factor-critical planar graphs, establishing a key property of their minimum degree.

Findings

Minimal k-factor-critical planar graphs have minimum degree k+1.

The conjecture by Favaron and Shi holds true for planar graphs.

Provides a proof confirming the conjecture in the planar case.

Abstract

A graph of order $n$ is said to be \emph{$k$-factor-critical} ($0\leq k <n$) if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph $G$ is \emph{minimal} if $G-e$ is not $k$-factor-critical for any edge $e$ in $G$. Favaron and Shi posed the conjecture that every minimal $k$-factor-critical graph is of minimum degree $k+1$ in 1998. In this paper, we confirm the conjecture for planar graphs.