## Graph Theory Lecture Notes14

### Vertex Coverings

**Def:** A *vertex covering* is a set of vertices in a graph such that every edge of the graph has at least
one end in the set. A *minimum covering* is a vertex covering which has the smallest number of
vertices for a given graph.
If M is a matching in a graph and K a covering of the same graph, then |M| <= |K|.

**Theorem**: If M is a matching, K a covering and |M| = |K|, then M is a maximum matching and K is
a minimum covering.

**Theorem**: (König) In a bipartite graph the number of edges in a maximum matching equals the
number of vertices in a minimum cover.

(Proof postponed)

*Alternate Proof of the Marriage Theorem:* If |N(S)| >= |S| for all S a subset of X, we need to show that there
exists an X-saturating matching. Since the graph is bipartite, and an X-saturating matching would be
a maximum matching, König's theorem implies that |K*| = |X| for a minimum covering K*, i.e., |X|
<= |K| for every cover K.

Let K be a covering, Z the set of X vertices in K, and S = X - Z. Since S is not in K, each of the
vertices of N(S) must be in K because K is a covering. Therefore, we have,

|K| >= |Z| + |N(S)| = |X| - |S| + |N(S)| >= |X| since |N(S)| >= |S|.
Thus, a minimum covering has at least |X| vertices, and by König's theorem a maximal matching has
at least |X| vertices, i.e., it must be X-saturating.
**Def**: In a 0,1 matrix, a set of 0 entries is called an* independent set of 0's* if no two 0's in the set are
in the same row or column. The rows and columns of the matrix are called *lines* in this context.

**Corollary** (König - Egerváry): In a 0,1 matrix the size of a maximum independent set of 0's equals
the minimum number of lines that cover all the 0's of the matrix.

*Pf:* Build a bipartite graph where the vertices of X are the rows of the matrix and the vertices of Y
are the columns. Join a vertex in X with one in Y if and only if there is a 0 in the intersection of
that row and column. Apply König's theorem to this bipartite graph.

### Edge Coverings

**Def**: A collection of edges in a graph such that every vertex is on one of these edges is called an
*edge covering* of the graph.