KU Math 725: Lecture 2/18/2026

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Matching and Covers

[definition=%counter%] A vertex cover of $G$ is a set $Q \subseteq V(G)$ that contains at least one endpoint of every edge.

An edge cover of $G$ is a set $L\subseteq E(G)$ that contains at least one edge incident to every vertex. [/definition]

[remark=%counter%] $V(G)$ is always a vertex cover. $E(G)$ is always an edge cover if an edge cover exists (one does not exist when there) [/remark]

[definition=%counter%] A matching on $G$ is an edge set $M\subseteq E(G)$ that includes at most one edge incident to each vertex.

A vertex is matched by $M$ if it is incident to an edge in $M$. The set of matched vertices is denoted $V(M)$. [/definition]

[remark=%counter%] Note that $|V(M)| = 2|M|$ since each edge contributes two vertices. [/remark]

[definition=%counter%] $M$ is maximal if it is not contained in any strictly larger matching.

$M$ is maximum if it has the largest possible size amongst all matchings. $M$ is perfect if $V(M) = V(G)$. [/definition]

[definition=%counter%] A $k$-factor in $G$ is a $k$-regular spanning subgraph. [/definition]

[remark=%counter%] A perfect matching is a $1$-factor. [/remark]

[remark=%counter%] The vertex analogue of a matching is a coclique (or indepedent set). [/remark]

[definition=%counter%] Notation:

  • $\alpha$ is the max size of a coclique.
  • $\alpha’$ is the max size of a matching.
  • $\beta$ is the min size of a vertex cover.
  • $\beta’$ is the min size of an edge cover. [/definition]

[remark=%counter%] Computing $\alpha’$ and $\beta’$ are equivalent and are in $P$.

Computing $\alpha$ and $\beta$ are equivalent are in $NP$.

ALl four are equivalent for bipartite graphs. [/remark]

[remark=%counter%] Counting maximal matchings is hard in general. There are formulas for $K_{2n}$ and $K_{n,n}$. There are no none formulas for $Q_n$. [/remark]

[example=%counter%] Compute $\alpha, \alpha’, \beta, \beta’$ for $C_n$.

  • $C_3$: $(1, 1, 2, 2)$
  • $C_4$: $(2, 2, 2, 2)$
  • $C_5$: $(2, 2, 3, 3)$

Might conjecture:

  • $\alpha = \alpha = \lfloor \frac{n}{2} \rfloor$
  • $\beta = \beta’ = \lceil \frac{n}{2} \rceil$

It is actually the case

\[\begin{align*} \alpha + \beta &= n, \\ \alpha' + \beta' &= n, \end{align*}\]

holds in general. [/example]

[proposition=%counter%] $\alpha + \beta = n$. [/proposition]

[proof] Let $Q$ be a subset of $V(G)$. It is apparent that $Q$ is a coclique if and only if $\overline{Q}$ is a vertex cover. So now make $Q$ of maximum size. Then $\overline{Q}$ is a vertex cover of minimum size. Thus, $\alpha + \beta = n$ (because $Q$ and $\overline{Q}$ are complements of each other). [/proof]