On the minimum degree of minimal $k$-$\{1,2\}$-factor critical $k$-planar graphs

TL;DR

This paper proves a conjecture about the minimum degree of minimal $k$-factor-critical graphs with $ ext{planar}$ properties, specifically for $k$-planar graphs, extending the understanding of factor-critical graphs in planar graph theory.

Contribution

It confirms the conjecture that minimal $k$-$ ext{ extless}1,2 ext{ extgreater}$-factor critical $k$-planar graphs have degrees between $k+1$ and $k+2$, including planar graphs.

Findings

Confirmed the degree bounds for $k$-planar graphs.

Extended the conjecture to planar graphs.

Resolved the conjecture for planar graphs.

Abstract

A graph of order $n$ is said to be $k$-\emph{factor-critical} $(0\le k<n)$ 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 $G$. In 1998, Favaron and Shi posed the conjecture that every minimal $k$-factor-critical graph is of minimum degree $k+1$. A natural extension of this notion arises from $\{1,2\}$-factors. A spanning subgraph of $G$ is called a $\{1,2\}$-factor if each of its components is a regular graph of degree one or two. A graph is $k$-\emph{$\{1,2\}$-factor critical} if the removal of any $k$ vertices results in a graph with a $\{1,2\}$-factor. A recent conjecture in the area states that every minimal $k$-$\{1,2\}$-factor critical graph $G$ satisfies $k+1\le \delta(G)\le k+2$. In this paper, we prove that the conjecture holds for $k$-planar graphs, that is, graphs in which the deletion of any set of $k$ vertices yields a planar graph. In particular, this resolves the conjecture for planar graphs.