A note on minimal matching covered graphs

A graph is called matching covered if for its every edge there is a maximum matching containing it. It is shown that minimal matching covered graphs contain a perfect matching.

💡 Research Summary

The paper investigates the relationship between two notions in matching theory: matching‑covered graphs and minimal matching‑covered graphs. A graph G is called matching‑covered if every edge belongs to at least one maximum matching of G. The subgraph C(G) is obtained by deleting all edges that never appear in any maximum matching; thus G is matching‑covered exactly when G = C(G). A minimal matching‑covered graph satisfies the additional condition that for every edge e, the graph G − e is not matching‑covered (i.e., (G − e) ≠ C(G − e)).

The author first establishes a lemma for connected matching‑covered graphs that do not possess a perfect matching. The lemma has two parts: (1) for any edge e = (u, v) there exists a maximum matching that misses either u or v; (2) distinct edges cannot have identical families of maximum matchings containing them (i.e., M(e) = M(e′) implies e = e′). The proof uses a distance‑based function μₑ(F) measuring how far the uncovered vertices of a matching F are from the endpoints of e, selects a matching minimizing this function, and derives a contradiction if both endpoints are covered. The second part constructs a new matching by swapping edges to show the families differ.

From the lemma, a corollary is derived for bipartite matching‑covered graphs: if one vertex on one side of the bipartition is missed by some maximum matching, then every vertex on that side can be missed by some maximum matching.

The main theorem states that any graph satisfying (i) it is matching‑covered and (ii) removal of any edge destroys the matching‑covered property must contain a perfect matching. The proof proceeds by assuming G is connected, picking an arbitrary edge e₀, and iteratively choosing edges e₁, e₂, … such that M(e_k) ⊇ M(e_{k+1}). Because the edge set is finite, some edge repeats: e_i = e_j with i < j. The construction guarantees M(e_i) = M(e_j) while e_{j‑1} ≠ e_j, contradicting part (2) of the lemma unless G already has a perfect matching. Hence a minimal matching‑covered graph is necessarily 1‑extendable (a connected matching‑covered graph containing a perfect matching).

The paper situates this result within the broader literature on graph cores and 1‑extendable graphs, referencing classic works by Dulmage‑Mendelsohn, Harary, Plummer, Lovász, and West. It concludes that minimal matching‑covered graphs are precisely the connected 1‑extendable graphs, thereby enriching the structural understanding of graphs where every edge is essential for the matching‑covered property.