KU Math 725: Lecture 2/20/2026
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Recall from last time:
[definition=%counter%]
- Coclique: set of vertices touching each edge at most once. $\alpha$ (max).
- Vertex cover: set of vertices touching each edge at least once. $\beta$ (min).
- Matching: a set of edges touching each vertex at most once. $\alpha’$ (max).
- Edge cover: a set of edges touching each vertex at least once. $\beta’$ (min). [/definition]
We also proved the following:
[proposition=%counter% (Proposition 3.5)]
\[\alpha + \beta = n\][/proposition]
Today we first show:
[proposition=%counter% (Gallai’s identity)]
\[\alpha' + \beta' = n.\][/proposition]
[proof] First show $\alpha’ + \beta’ \le n$. Let $M$ be a max matching, so $|M| = \alpha’$, $A$ be a collection of edges, each incident to one vertex that is not matched by $M$; these edges are all distinct (otherwise $M$ is not max). Then
\[|A| = n - |V(M)| = 2 - \alpha' = n - 2|M|.\]On the other hand $A\cup M$ is an edge cover, so
\[|A \cup B| \ge \beta'.\]So
\[\alpha' + \beta' \le |M| + |A\cup M| \le 2|M| + |A| = n.\]Now we show $\alpha’ + \beta’ \ge n$. Let $L$ be a min edge cover, so $|L| = \beta’$. For every edge $e = xy \in L$, we must have $\deg_L(x)=1$ or $\deg_L(y) = 1$, otherwise $e$ could be removed from $L$ to yield a smaller edge cover. This implies that every component of $L$ is a star. This implies that $L$ is acyclic. So then
\[|L| = n - c(L).\]We construct a matching $M$ by choose one edge from every component of $L$. Then
\[|M| = c(L) \le \alpha'.\]Thus,
\[\alpha' + \beta' \ge |M| + |L| = n.\][/proof]
[proposition=%counter% (3.7)]
\[\alpha' \le \beta\][/proposition]
[proof] If $M$ is a matching in $G$, then no vertex can cover more than one of the edges in $M$. So every vertex cover must have at least $|M|$ vertices. [/proof]
[remark=%counter%] This kind of result is known as weak duality. Every matching has size less or equal to the size of every vertex cover.
This implies that if we find a matching and a vertex cover of the same size, then the matching is maximum and the vertex cover is minimum. Such a pair may not always exist. [/remark]
[theorem=%counter% (Konig-Egervery)] If $G$ is bipartite, then $\alpha’ = \beta$. [/theorem]
[definition=%counter%] Let $M$ be a matching in $G$ and $P$ a path in $G$. A path $P$ is $M$-alternating if its edges alternate between edges in $M$ and edges not in $M$. $P$ is $M$-augmenting if it is $M$-alternating and both endpoints are unmatched by $M$. [/definition]
[remark=%counter%] Every $M$-augment path has an even number of vertices. (Two unmatched endpoints and an even number of matched interior vertices. Hence it has odd length.) [/remark]
[remark=%counter%] If $P$ is an $M$-augmenting path, then
\[M \Delta P = (M \cup P) \setminus (M \cap P).\]is a matching with one more edge than $M$.
Fact 1: If $C = A \Delta B$, then $A = B \Delta C$ and $B = A \Delta C$. This says that every element is contained in an even number of the sets $A, B, C$. [/remark]
[lemma=%counter% (Lemma 3.10)] Let $M$, $N$ be matchings in $G$. Then every nontrivial component of $M \Delta N$ is either a path or an even cycle. [/lemma]
[proof] Each vertex of $G$ can have degree at most $2$ in $M \Delta N$. Thus every nontrivial component is either a path or a cycle. If it is a cycle, then its edges must alternate between $M$ and $N$, so it must be even. [/proof]
[theorem=%counter% (Berge, 1957)] Let $M$ be a matching on $G$. Then $M$ is a matching if and only if $G$ contains no $M$-augmenting path. [/theorem]
($\implies$) Obvious.
($\impliedby$) Next time!