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.

• The paper proves that every minimal matching covered graph with minimum degree 3 contains at least two nonadjacent 3-lines and four cubic vertices.
• It employs tight cut theory and ear decomposition techniques to analyze degree properties and ensure structural constraints through inductive arguments.
• The results generalize classical bipartite outcomes to nonbipartite cases, offering explicit constructions that confirm the optimality of the derived bounds.

Adjacent Vertices of Small Degree in Minimal Matching Covered Graphs

Introduction and Context

The paper addresses the structural properties of minimal matching covered graphs, focusing on the local degree properties of their vertices—specifically, the existence of pairs of adjacent vertices of low degree. Minimal matching covered graphs (also called minimal matching covered or minimal elementary graphs) are those in which every edge is contained in a perfect matching and the deletion of any edge destroys this property. These graphs are central to matching theory, as they provide fine-grained insight into permissible configurations supporting perfect matchings.

The study is motivated in part by foundational results for bipartite graphs—most notably, Lovász and Plummer’s theorem [J. Combin. Theory Ser. B, 1977] that guarantees many 2-lines (edges joining two degree-2 vertices) in minimal matching covered bipartite graphs, and by more recent work showing that the minimum degree in a minimal matching covered graph is always 2 or 3. The present paper generalizes and sharpens these results, particularly outside the bipartite case, and investigates the existence and distribution of edges joining vertices of minimal or near-minimal degree (so-called 2-lines and 3-lines).

Main Contributions

The central results of the paper are:

• Theorem 1: Every minimal matching covered graph with at least 4 vertices and minimum degree at least 3 contains at least two nonadjacent 3-lines (edges joining two cubic—degree 3—vertices), and consequently at least four cubic vertices. These lower bounds are shown to be tight.
• Theorem 2: Every minimal matching covered graph (with at least 4 vertices) contains two nonadjacent edges, each being a 2-line or 3-line.
• Sharpness of these results is demonstrated via explicit constructions.
• The results generalize the state-of-the-art from bipartite to nonbipartite cases, and clarify the landscape of local-degree constraints in the minimal setting.

Technical Framework

The authors employ a structural decomposition approach rooted in classical tight cut theory and ear decompositions. Key technical tools include:

• Removable Edges and Classes: A central notion is whether an edge is removable (i.e., its deletion preserves matching coveredness); minimal graphs are precisely those with no removable edges.
• Tight Cuts and Contractions: The use of tight cuts, their contractions, and the interplay with matching coveredness is essential. Leveraging Lovász's result that contracting tight cuts of a matching covered graph yields matching covered subgraphs is critical for induction and minimal counterexample arguments.
• Barriers and Bicriticality: The barrier structure characterization (barriers are sets whose removal induces as many odd components as their size) and the distinction between bicriticality and other minimal matching covered graphs (according to Lovász and Plummer) are central.

To analyze the occurrence of low-degree vertices and lines, the authors:

• Develop inductive arguments that repeatedly contract tight cuts, isolate subcomponents, and track degree and minimality constraints into these subcomponents.
• Extensively use the concept of $k$-lines, watching their evolution under contraction and decomposing cycles and forests induced by nonremovable edges, particularly in the bipartite case.
• Apply extremal arguments, showing that the lower bounds on cubic vertices and 3-lines are realized.

Numerical and Structural Bounds

The paper’s main theorems provide tight lower bounds in the form of integer constraints:

• In the minimum degree 3 case, any minimal matching covered graph with $n \geq 4$ vertices must have at least four cubic vertices and, correspondingly, at least two nonadjacent 3-lines.
• This result is tight: the authors construct an infinite family, $G_n$, achieving exactly four cubic vertices and two nonadjacent 3-lines.
• For arbitrary minimum degree (2 or 3), every such graph must contain two nonadjacent edges each being a 2-line or 3-line. This covers the general (possibly mixed degree) situation.

The authors also prove that the lower quantity of cubic vertices and 3-lines in the main theorems cannot be improved for any graph of size greater than 6.

Implications and Future Directions

The work settles several open questions regarding the local degree structure of minimal matching covered graphs:

• For extremal graph theory, these results delineate minimal overlap (adjacency) among low-degree vertices in the minimal perfect matching context.
• The findings have implications for the theory of brick and brace decomposition since minimal matching covered graphs are constructed from these building blocks via tight cut decompositions.
• They highlight the subtle differences between the bipartite and nonbipartite cases, providing essential insight for the classification and enumeration of such graphs.
• The explicit construction showing sharpness may inform further algorithmic and enumerative investigations.

Lastly, the authors formulate a conjecture regarding minimal braces (bipartite analogs of bricks), proposing that every minimal brace with at least 6 vertices contains a 3-line. This points to ongoing structural questions in the interaction between minimality, perfect matchings, and local degree.

Conclusion

This paper advances the theory of minimal matching covered graphs by establishing precise, sharp lower bounds for the presence of low-degree adjacency (2-lines and 3-lines), unifying and extending previous results across the bipartite and nonbipartite divide. Through technical arguments relying on tight cut contractions, barriers, and inductive analysis, the authors map the landscape of possible degree configurations, closing several gaps and highlighting new conjectures for future inquiry. The results are expected to inform both deeper combinatorial theory and practical approaches to matching structure recognition and decomposition.