The minimum degree of minimal 2-extendable claw-free graphs

TL;DR

The paper investigates the minimum degree of minimal $2$-extendable claw-free graphs, building on prior results for general and $1$-extendable cases. It leverages the Tutte-type minimal $k$-extendable graph characterization of Anunchuen and Caccetta and a detailed analysis of edge-termination structures, including $5$-vertex cuts and odd components, to rule out the possibility of $\delta(G)\ge 6$. The main finding is that a minimal $2$-extendable claw-free graph must satisfy $\delta(G)\in\{4,5\}$, thereby refining the understanding of extendability within claw-free graphs and aligning with a broader conjectured pattern for minimal $k$-extendable claw-free graphs. The results contribute to the structural theory of matchings in claw-free graphs and may inform related decomposition and extremal results in matching theory.

Abstract

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal if the deletion of any edge results in a graph that is not $k$-extendable. In 1994, Plummer proved that every $k$-extendable claw-free graph has minimum degree at least $2k$. Recently, He et al. showed that every minimal 1-extendable graph has minimum degree 2 or 3. In this paper, we prove that the minimum degree of a minimal 2-extendable claw-free graph is either $4$ or $5$.

Figures (8)

• Figure 1: All the black (resp. white) vertices in $S_e$ form the independent set $A$ (resp. $B$).
• Figure 2:
• Figure 3: (a) $|V(O_1)|>1$ and $|V(O_2)|=1$; (b) $|V(O_1)|>1$ and $|V(O_2)|>1$.
• Figure 4: (a) $e$ is of type 1 if $|S_e|=4$; (b) $e$ is of type 2 if $|S_e|=5$.
• Figure 5: (a) $X_{e}^{u}$ satisfies Property $P$; (b) $X_{e_1}^{u}$ satisfies Property $P$.

Theorems & Definitions (35)

• Theorem 1.1: P
• Theorem 1.2: P
• Theorem 1.3: AC1997
• Theorem 1.4: PL
• Theorem 1.5
• Theorem 2.1: AC1994
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4