Adjacent vertices of small degree in minimal matching covered graphs

TL;DR

This paper investigates the structure of minimal matching covered graphs, establishing new lower bounds on the number of specific edges and vertices with degree 2 or 3, refining previous results.

Contribution

It proves that such graphs with at least 4 vertices contain at least two nonadjacent edges that are 2-lines or 3-lines, and at least four degree-3 vertices when minimum degree is 3.

Findings

Every minimal matching covered graph with at least 4 vertices has two nonadjacent edges that are 2-lines or 3-lines.

Minimal matching covered graphs with minimum degree 3 have at least four vertices of degree 3.

The bounds for the number of 3-lines and cubic vertices are shown to be sharp.

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 $k$. Lov\'asz 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.