Adjacent vertices of small degree in minimal matching covered graphs

Abstract

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An edge is called a $k$-line if both of its end vertices are of degree $k$. Lovász and Plummer [J. Combin. Theory, Ser. B 23 (1977) 127--138] proved that a minimal matching covered bipartite graph different from $K_2$ has minimum degree 2 and contains at least $[(|V(G)|+15)/6]$ 2-lines by ear decompositions. He et al. [J. Graph Theory 111 (2026) 5--16] showed that the minimum degree of a minimal matching covered graph different from $K_2$ is either 2 or 3. In this paper, we prove that every minimal matching covered graph with at least 4 vertices contains at least two nonadjacent edges, each of which is either a 2-line or a 3-line. Consequently, we show that every minimal matching covered graph with at least 4 vertices and minimum degree 3 contains at least 4 vertices of degree 3. Furthermore, the lower bounds for both the number of 3-lines and the number of cubic vertices are sharp.

Figures (6)

• Figure 1: The graph $G_{5}$ (the bold edges are 3-lines).
• Figure 2: The graphs $C_4^+$ (left), and $H_5$ (right).
• Figure 3: Illustration for the proof of Lemma \ref{['lem:nonre-forest']}.
• Figure 4: The graph $R_8$.
• Figure 5: Illustration for the proof of Lemma \ref{['lem:WL-bicritical']}, where dashed lines indicate the possibility of an edge.

Theorems & Definitions (41)

• Theorem 1.1: LP77
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1: Lovasz87
• Theorem 2.2: Tutte47
• Corollary 2.3: Lovasz87
• Proposition 2.4: LP86
• Corollary 2.5