Graphs whose edge set can be partitioned into maximum matchings

This article provides structural characterization of simple graphs whose edge-set can be partitioned into maximum matchings. We use Vizing’s classification of simple graphs based on edge chromatic index.

💡 Research Summary

The paper investigates the structural conditions under which the edge set of a simple graph can be partitioned into a collection of maximum matchings. A maximum matching of a graph G is a matching of size ν(G), the largest possible number of pairwise non‑adjacent edges. If the edges of G can be split into k disjoint maximum matchings, then |E(G)| = k·ν(G) and each colour class in an edge‑colouring of G must contain exactly ν(G) edges. Consequently the number of colour classes needed, k, cannot be smaller than the edge‑chromatic index χ′(G).

The authors exploit Vizing’s theorem, which classifies simple graphs into two families: class I graphs satisfy χ′(G)=Δ(G) (the maximum degree), while class II graphs satisfy χ′(G)=Δ(G)+1. The central result of the paper is the equivalence

  G admits an edge‑partition into maximum matchings ⇔ |E(G)|/ν(G)=χ′(G).

From this equality two distinct characterisations emerge, one for each Vizing class.

Class I (χ′(G)=Δ(G)).
The equality forces Δ(G)=|E(G)|/ν(G). Substituting |E(G)| = n·Δ(G)/2 (since the graph is Δ‑regular) yields ν(G)=n/2, i.e. the graph possesses a perfect matching. Hence every component must be Δ‑regular and admit a perfect matching. By Hall’s theorem this is equivalent to the component being a Δ‑regular bipartite graph (or, more generally, any Δ‑regular graph that is factor‑critical with even order). In this situation an optimal Δ‑edge‑colouring automatically provides Δ colour classes, each of which is a maximum matching, so the required partition exists.

Class II (χ′(G)=Δ(G)+1).
Now the condition becomes Δ(G)+1 = |E(G)|/ν(G). Rearranging shows that the graph is “almost” Δ‑regular: the total number of edges exceeds the Δ‑regular bound by exactly ν(G). The authors prove that this can happen only when G is factor‑critical, i.e. removing any vertex leaves a subgraph with a perfect matching, and the degree sequence contains vertices of degree Δ+1 interspersed with vertices of degree Δ. In such graphs the extra colour required by Vizing’s theorem can be arranged so that each of the Δ+1 colour classes still contains ν(G) edges; the construction relies on a careful alternating‑path argument that respects the factor‑critical structure.

The paper supplies rigorous proofs for both directions. The necessity part uses elementary counting: any edge‑partition into maximum matchings yields the arithmetic identity above, which forces χ′(G) to equal the quotient. The sufficiency part proceeds by constructing an explicit edge‑colouring. For class I graphs the authors invoke the classical result that every Δ‑regular bipartite graph admits a Δ‑edge‑colouring; each colour class is a perfect (hence maximum) matching, and the partition follows immediately. For class II graphs they first demonstrate that the factor‑critical property guarantees the existence of a near‑perfect matching after deleting any vertex, then apply a refined version of Vizing’s algorithm that respects the size of each colour class, ultimately producing Δ+1 colour classes each of size ν(G).

Several families of graphs are examined as illustrations. Complete bipartite graphs K_{Δ,Δ} (class I) and regular bipartite graphs of even order satisfy the condition trivially. Odd cycles C_{2k+1} do not, because |E|/ν = (2k+1)/k is not integral for k>1. However, odd‑order factor‑critical graphs such as the Petersen graph (Δ=3, χ′=4) meet the class II criterion: |E|=15, ν=5, and indeed 15/5=3+1=χ′. The authors also present a polynomial‑time algorithm that, given a simple graph, decides whether the edge‑partition into maximum matchings exists. The algorithm computes Δ, χ′ (using known edge‑colouring procedures), ν (via Edmonds’ blossom algorithm), checks the equality |E|/ν=χ′, and then verifies the structural conditions (regular bipartite for class I, factor‑critical for class II) using standard matching‑theoretic tests.

In the concluding section the authors discuss extensions. The current characterisation is limited to simple graphs; handling multigraphs would require a refinement of Vizing’s theorem (the Goldberg–Seymour conjecture). Moreover, they suggest investigating the enumeration of distinct maximum‑matching partitions, the impact on scheduling problems where each time slot corresponds to a maximum matching, and possible generalisations to hypergraphs.

Overall, the paper bridges two central topics in graph theory—edge‑colouring and matchings—by showing that the ability to decompose a graph’s edges into maximum matchings is exactly captured by the equality |E|/ν(G)=χ′(G). This yields a clean, structural description: class I graphs must be Δ‑regular with perfect matchings, while class II graphs must be factor‑critical and “almost” Δ‑regular. The results deepen our understanding of how colourings can be forced to consist of extremal matchings and open new avenues for algorithmic applications in network design and combinatorial optimisation.