On minimal k-factor-critical planar graphs

TL;DR

This work addresses Favaron and Shi's conjecture that minimal $k$-factor-critical graphs have minimum degree $k+1$, focusing on planar graphs. The authors establish the $k=3$ case by a vertex-cut and minor-avoidance argument that excludes $\delta(G)\ge5$ and hence forces $\delta(G)=4$ for minimal 3-fc planar graphs, then reconcile with known results for $k\in\{1,2\}$ and the planar bound $\delta(G)\le5$ to conclude that planar minimal $k$-fc graphs satisfy $\delta(G)=k+1$ (with $k\le4$). The results integrate matching theory with planarity constraints to advance the understanding of minimal fc graphs and suggest techniques applicable to broader graph classes.

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.

Figures (3)

• Figure 1: Illustration for the proof of Lemma \ref{['propertyP']}.
• Figure 2: Illustration for a partition of $V(G)$ (the rectangles in the figure mean the vertex sets).
• Figure 3: The illustration for a $K_{3,3}$ minor.

Theorems & Definitions (15)

• Lemma 1.1: O.Favaron
• Conjecture 1.2: Shizh
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: KuratowskiWagner
• Theorem 2.4: tut
• Lemma 2.5
• Lemma 2.6
• proof