Lecture Roadmap
Topics for Today
- The order of a graph and the main parameters we will measure against it.
- The basic inequality between a matching and a vertex cover.
- Konig's theorem: in a bipartite graph, maximum matching size equals minimum vertex cover size.
- The meaning of a min-max theorem and some standard examples.
- Gallai's identity relating independent sets and vertex covers.
- A basic upper bound for domination number in terms of minimum degree.
The Unifying Theme
Much of this lecture is about a recurring graph-theoretic tension:
Matchings pack disjoint edges. Vertex covers hit every edge. Independent sets avoid every edge. Dominating sets cover vertices through adjacency. Once the parameters are named carefully, many proofs become short and conceptual.
Parameters to Track
Order of a Graph and Why It Matters
Definitions
If \(G=(V,E)\) is a finite graph, then its order is the number of vertices:
Many books simply write \(n\), and some older texts use \(p\). The number of edges is called the size of the graph and is written \(m(G)=|E(G)|\) or just \(m\).
Handshake Lemma
The degree sum equals twice the number of edges:
This is often the first way order and size interact. It lets us pass from a local condition on degrees to a global statement about the graph.
Why Order Appears in This Lecture
- Gallai's identity has the form \(\alpha(G)+\tau(G)=n(G)\).
- Domination bounds are measured against the total number of vertices.
- Min-max theorems often say a packing number plus a complementary covering number is controlled by \(n\).
Examples
| Graph | Order \(n\) | Size \(m\) | Comment |
|---|---|---|---|
| \(P_5\) | 5 | 4 | A path on five vertices. |
| \(C_6\) | 6 | 6 | A cycle has order equal to its size. |
| \(K_{3,4}\) | 7 | 12 | Complete bipartite graph with \(3 \cdot 4\) edges. |
Relationship Between Matching and Vertex Cover
Definitions
A vertex cover of \(G\) is a set \(C \subseteq V(G)\) such that every edge of \(G\) has at least one endpoint in \(C\). The minimum size of a vertex cover is denoted by \(\tau(G)\).
A maximum matching has size \(\nu(G)\).
Basic Inequality
For every graph \(G\),
Proof
Let \(M\) be any matching and let \(C\) be any vertex cover. Every edge of \(M\) must be covered by some vertex of \(C\).
Because the edges of a matching are pairwise disjoint, two different edges of \(M\) cannot be covered by the same endpoint. So \(C\) must contain at least one distinct vertex for each edge of \(M\).
Now maximize over all matchings and minimize over all vertex covers. This gives \(\nu(G) \le \tau(G)\).
What the Inequality Means
A matching is a packing problem: we try to place as many pairwise disjoint edges as possible.
A vertex cover is a covering problem: we try to hit every edge using as few vertices as possible.
The inequality \(\nu(G) \le \tau(G)\) is the obvious direction in the packing-versus-covering story.
Equality Need Not Hold in General
Take the triangle \(K_3\).
- A maximum matching has size \(1\), so \(\nu(K_3)=1\).
- A minimum vertex cover has size \(2\), so \(\tau(K_3)=2\).
Thus equality fails for non-bipartite graphs. The bipartite case is special.
Bipartite Graphs: Maximum Matching Equals Minimum Vertex Cover
Konig's Theorem
If \(G=(X \cup Y,E)\) is bipartite, then
Proof Strategy
We already know \(\nu(G)\le\tau(G)\). So the whole job is to start from a maximum matching and build a vertex cover of the same size.
The construction uses alternating paths, exactly the same language that appears in matching theory more generally.
Alternating-Path Construction
Let \(M\) be a maximum matching in the bipartite graph \(G=(X \cup Y,E)\). Let \(U \subseteq X\) be the set of vertices of \(X\) not saturated by \(M\).
Starting from the vertices of \(U\), follow alternating paths that begin with an edge not in \(M\), then an edge in \(M\), and so on. Let \(Z\) be the set of all vertices reachable this way. Write
Now define
We claim that \(C\) is a vertex cover and that \(|C|=|M|\).
Why \(C\) Covers Every Edge
Take any edge \(xy\) with \(x \in X\) and \(y \in Y\). Suppose neither endpoint lies in \(C\). Then \(x \in Z_X\) and \(y \notin Z_Y\).
If \(xy \notin M\), then an alternating path reaching \(x\) can be extended across \(xy\), so \(y\) would lie in \(Z_Y\), contradiction.
If \(xy \in M\), then whenever \(x \in Z_X\), its matched partner must lie in \(Z_Y\). Again we get \(y \in Z_Y\), contradiction.
So every edge has at least one endpoint in \(C\), which means \(C\) is a vertex cover.
Why \(|C|=|M|\)
No vertex of \(Z_Y\) is unmatched, because then an alternating path from \(U\) to that unmatched vertex would be an augmenting path, contradicting the assumption that \(M\) is maximum.
Each matched edge of \(M\) contributes exactly one vertex to \(C\):
- If its \(X\)-endpoint lies in \(Z_X\), then its \(Y\)-endpoint lies in \(Z_Y\), so the cover takes the \(Y\)-endpoint.
- If its \(X\)-endpoint does not lie in \(Z_X\), then its \(Y\)-endpoint does not lie in \(Z_Y\), so the cover takes the \(X\)-endpoint.
Unmatched vertices of \(X\) lie in \(Z_X\), so they are not counted in \(C\). Hence \(C\) contains exactly one endpoint from each edge of \(M\), and therefore
Conclusion
Since \(C\) is a vertex cover, \(\tau(G)\le |C|=|M|=\nu(G)\). Combined with the general inequality \(\nu(G)\le \tau(G)\), we obtain
Min-Max Theorems and Examples
General Idea
A min-max theorem says that the largest size of one kind of object equals the smallest size of another kind of object that blocks it.
Usually the easy part is the inequality \(\max \le \min\). The real work is proving equality by building a witness on the minimizing side from an optimal maximizing object, or vice versa.
Examples
| Theorem | Maximum Side | Minimum Side |
|---|---|---|
| Konig | Maximum matching in a bipartite graph | Minimum vertex cover |
| Max-flow min-cut | Largest feasible \(s\)-\(t\) flow value | Smallest \(s\)-\(t\) cut capacity |
| Menger (edge version) | Maximum number of edge-disjoint \(s\)-\(t\) paths | Minimum size of an \(s\)-\(t\) edge cut |
| Menger (vertex version) | Maximum number of internally disjoint \(u\)-\(v\) paths | Minimum number of vertices separating \(u\) and \(v\) |
A Useful Mental Template
- Name the packing parameter.
- Name the covering parameter.
- Prove the obvious inequality \(\max \le \min\).
- Find a structure that turns an optimal object on one side into an equally good object on the other.
Konig's theorem is the cleanest example of this template in elementary matching theory.
Gallai's Theorem
Gallai's Identity
For every graph \(G\) of order \(n\),
where \(\alpha(G)\) is the size of a maximum independent set and \(\tau(G)\) is the size of a minimum vertex cover.
Proof
Let \(I\) be an independent set. Then no edge has both endpoints in \(I\). Therefore every edge has at least one endpoint in \(V(G)\setminus I\), so \(V(G)\setminus I\) is a vertex cover.
This shows that for every independent set \(I\),
Taking the largest possible independent set gives
Conversely, let \(C\) be a vertex cover. If two vertices outside \(C\) were adjacent, that edge would have no endpoint in \(C\), impossible. Hence \(V(G)\setminus C\) is an independent set, so
The two inequalities combine to give \(\alpha(G)=n-\tau(G)\), or equivalently
Interpretation
Independent sets and vertex covers are complements of one another. Gallai's theorem makes that precise at the level of optimum sizes.
So the order \(n\) of the graph is not just background notation. It is the total amount of vertex mass split between an optimal sparse set and an optimal covering set.
Bipartite Consequence
In a bipartite graph, Konig plus Gallai gives
So once we know the maximum matching size in a bipartite graph, we immediately know its minimum vertex cover size and maximum independent set size.
Remark on Terminology
Some texts also attach Gallai's name to the edge-cover identity \(\nu(G)+\rho(G)=n\) for graphs without isolated vertices. In this lecture we are using the independent-set and vertex-cover form.
Dominating Set Size and Minimum Degree
Definition
A set \(D \subseteq V(G)\) is a dominating set if every vertex of \(G\) either belongs to \(D\) or is adjacent to a vertex of \(D\). The minimum size of a dominating set is the domination number, denoted \(\gamma(G)\).
A Basic Degree Bound
If \(G\) has order \(n\) and minimum degree \(\delta(G)=\delta\), then
Proof
Choose any vertex \(v\). Let \(N(v)\) be its open neighborhood, and consider the set
The set \(D\) contains the vertex \(v\) itself. Every vertex in \(N(v)\) is adjacent to \(v\), so it is dominated by \(D\). Every vertex outside \(N(v)\) already lies in \(D\). Hence \(D\) is a dominating set.
Therefore
Since \(\deg(v)\ge \delta\), we get
Quick Example
For the complete graph \(K_n\), we have \(\delta=n-1\). The bound gives
which is exact, since any single vertex dominates the whole graph.
How Sharp Is This?
The estimate \(\gamma(G)\le n-\delta\) is easy and useful, but it is often not best possible. For example, when \(\delta \ge 1\), a stronger theorem of Ore gives \(\gamma(G)\le n/2\).
Still, the present bound is a good first illustration of how local degree information forces a global covering property.
Takeaways
Main Results
- The order of a graph is \(n=|V(G)|\), and many optimization identities are measured against it.
- For every graph, \(\nu(G)\le \tau(G)\).
- For every bipartite graph, Konig's theorem gives \(\nu(G)=\tau(G)\).
- Gallai's identity gives \(\alpha(G)+\tau(G)=n\).
- A graph of order \(n\) and minimum degree \(\delta\) satisfies \(\gamma(G)\le n-\delta\).
Proof Patterns to Remember
- Compare a packing and a cover by observing that each packed object needs a distinct witness in any cover.
- In bipartite graphs, alternating paths convert a maximum matching into a minimum vertex cover.
- Complementation turns independent sets into vertex covers and vice versa.
- To build a dominating set, sometimes it is easier to keep one useful vertex and delete its neighborhood.