On the minimum degree of minimal $k$-$\{1,2\}$-factor critical $k$-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 $k+1\le δ(G)\le k+2$. 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.

Figures (5)

• Figure 1: Example of a non-planar graph
• Figure 2: Example of a non-planar and $1$-planar graph
• Figure 3: Planar $k$-$\{1,2\}$-factor-critical graphs with $k=1$. In each example, $G-e$ is not a $k$-$\{1,2\}$-factor-critical graph and $\delta(G)=k+2=3$ for the graph on the left, while $\delta(G)=k+1=2$ for the graph on the right. The highlighted vertices are sets $S\subset V(G)$ satisfying $i(G-e-S)>\left|S\right|-k$, which verifies that $G-e$ is not a $k$-$\{1,2\}$-factor-critical graph by \ref{['asd12392']}.
• Figure 4: Planar $0$-$\{1,2\}$-factor-critical graph with minimum degree $\delta(G)=0+3=3$. Moreover, $G-e$ is not a $0$-$\{1,2\}$-factor-critical graph, which can be verified using the highlighted set via \ref{['asd12392']}.
• Figure 5: The graph in \ref{['1231ss12']}, \ref{['1231szzzs12']} is a $2$-$\{1,2\}$-factor-critical ($2$-factor-critical) graph.

Theorems & Definitions (32)

• Theorem 1.1: tutte19531
• Theorem 1.2: tutte1947factorization
• Theorem 1.3: favaron1996k
• Theorem 1.4: ksachscritical
• Conjecture 1.5: favaron1996minimallyzhang2010equivalence
• Conjecture 1.7: KEVINksachsminimal1
• Theorem 1.8: KEVINksachsminimal1
• Theorem 1.9: KEVINksachsminimal1
• Theorem 3.1: kuratowski1930problemewagner1937eigenschaft
• Lemma 3.2: KEVINcuatriupla