Lecture Roadmap
Topics for Today
- Wagner's theorem and its proof through forbidden minors.
- Euler's formula and the most common corollaries for edge bounds and degrees.
- Duality of plane graphs and the cycle-cut correspondence.
- The planar separator theorem and the main idea of the proof.
- The 5-color theorem and its standard inductive proof with Kempe chains.
- The 4-color theorem and the ideas behind the computer-assisted proof.
- Straight-line embeddings of planar graphs.
- How planarity is tested in theory and in algorithms.
The Unifying Theme
Planar graphs are special for three different reasons at once.
Euler's formula gives quantitative control. Duality converts faces into vertices and cycles into cuts. Separator theorems explain why many hard problems become easier on planar inputs. Coloring and embedding results show that the geometry of the plane is restrictive but also constructive.
Vocabulary We Will Use
Plane Versus Planar
A plane graph is an actual drawing in the plane with no edge crossings. A planar graph is an abstract graph for which at least one such drawing exists.
Duality depends on the embedding, so it belongs to the language of plane graphs. Wagner's theorem, separator theorems, coloring, and planarity testing are statements about planar graphs as abstract objects.
Euler's Formula and Its Common Corollaries
Euler's Formula
If \(G\) is a connected plane graph with \(n\) vertices, \(m\) edges, and \(f\) faces, then
Proof
If \(G\) has a cycle, delete one edge from that cycle. The graph stays connected, the number of vertices stays the same, the number of edges drops by \(1\), and the two faces on either side of the deleted edge merge into one, so the number of faces also drops by \(1\). Therefore \(n-m+f\) is unchanged.
Repeat until no cycle remains. What remains is a tree \(T\) on the same vertex set. A tree has \(m(T)=n(T)-1\), and a connected plane tree has exactly one face, namely the unbounded face. So
Since the quantity stayed invariant during every deletion, the original connected plane graph also satisfies \(n-m+f=2\).
Most Used Corollaries
| Hypothesis | Conclusion | Why It Matters |
|---|---|---|
| Simple planar, \(n \ge 3\) | \(m \le 3n-6\) | Planar graphs are sparse. |
| Simple bipartite planar, \(n \ge 3\) | \(m \le 2n-4\) | Rules out \(K_{3,3}\). |
| Simple planar | Average degree \(<6\) | There exists a vertex of degree at most \(5\). |
| Simple bipartite planar | Average degree \(<4\) | Much stronger sparsity in the bipartite case. |
| Simple planar | Graph is \(5\)-degenerate | Starts inductive coloring arguments. |
Deriving \(m \le 3n-6\)
In a simple plane graph every face has length at least \(3\). If we sum face lengths, each edge is counted twice, so
Using Euler's formula \(f=2-n+m\), we obtain
Deriving \(m \le 2n-4\) for Bipartite Planar Graphs
A bipartite graph has no odd cycle, so in a plane embedding every face has length at least \(4\). Hence
Substitute \(f=2-n+m\):
This immediately shows that \(K_{3,3}\) is non-planar because it has \(n=6\) and \(m=9>8\).
Degree Consequences
From \(m \le 3n-6\) we get
Therefore some vertex has degree at most \(5\). Repeating this argument on every subgraph shows that every planar graph is \(5\)-degenerate. That tiny counting fact powers the 5-color theorem and many planar graph algorithms.
Wagner's Theorem and Forbidden Minors
Definitions
A graph \(H\) is a minor of \(G\) if it can be obtained from a subgraph of \(G\) by repeatedly contracting edges. A subdivision of a graph is obtained by replacing edges by internally disjoint paths.
Kuratowski's theorem is phrased with subdivisions. Wagner's theorem is phrased with minors.
Wagner's Theorem
A finite graph \(G\) is planar if and only if it contains neither \(K_5\) nor \(K_{3,3}\) as a minor.
Planar Graphs Are Closed Under Taking Minors
Deleting a vertex or edge from a planar drawing clearly preserves planarity. Contracting an edge in a planar drawing can be done inside a tiny neighborhood of that edge, so no crossing is introduced.
Hence every minor of a planar graph is planar.
Why \(K_5\) and \(K_{3,3}\) Are Not Planar
\(K_5\) has \(n=5\) and \(m=10\), but a simple planar graph on five vertices can have at most \(3n-6=9\) edges.
\(K_{3,3}\) is bipartite with \(n=6\) and \(m=9\), but a simple bipartite planar graph on six vertices can have at most \(2n-4=8\) edges.
Proof of Wagner's Theorem
If \(G\) is planar, then by the closure statement above every minor of \(G\) is also planar. Since neither \(K_5\) nor \(K_{3,3}\) is planar, \(G\) cannot contain either as a minor.
Conversely, assume \(G\) is non-planar. By Kuratowski's theorem, \(G\) contains a subgraph \(H\) that is a subdivision of \(K_5\) or of \(K_{3,3}\). Contract each subdivided path in \(H\) to a single edge. The result is exactly \(K_5\) or \(K_{3,3}\). Therefore \(G\) contains \(K_5\) or \(K_{3,3}\) as a minor.
So \(G\) is planar exactly when these two forbidden minors do not appear.
Kuratowski Versus Wagner
| Theorem | Obstructions | Operation Allowed |
|---|---|---|
| Kuratowski | \(K_5\), \(K_{3,3}\) | Subdivisions |
| Wagner | \(K_5\), \(K_{3,3}\) | Minors |
The two theorems are equivalent in strength. Kuratowski is more geometric; Wagner fits better with the language of structural graph theory.
Duality of Plane Graphs
Definition of the Dual
Let \(G\) be a connected plane graph. Put one vertex in each face of \(G\). For every edge \(e\) of \(G\), draw an edge \(e^*\) between the two face-vertices on either side of \(e\), crossing \(e\) exactly once. The resulting plane graph is the dual graph \(G^*\).
Bridges in \(G\) become loops in \(G^*\), and parallel edges may appear even when \(G\) is simple.
Basic Identities
For a connected plane graph \(G\),
In particular, Euler's formula for \(G\) and for \(G^*\) are the same statement with vertices and faces interchanged.
The Dual Depends on the Embedding
If a planar graph has two non-equivalent plane embeddings, their dual graphs need not be isomorphic.
So duality is a property of a plane graph, not only of the underlying abstract planar graph.
Cycle-Cut Duality
Let \(X \subseteq E(G)\). Then \(X\) is the edge set of a cycle in \(G\) if and only if \(X^*=\{e^*: e \in X\}\) is a bond, that is, a minimal cut, in \(G^*\). Dually, bonds in \(G\) correspond to cycles in \(G^*\).
A cycle in the plane separates inside from outside. Every dual edge crossing that cycle goes from one side to the other, so deleting those dual edges disconnects the dual. Minimality comes from the fact that each edge on the cycle is needed to keep the two sides separated. Reversing the argument gives the converse.
Spanning Trees and Dual Trees
If \(T\) is a spanning tree of a connected plane graph \(G\), then the set of dual edges corresponding to \(E(G)\setminus E(T)\) forms a spanning tree of \(G^*\).
Why? Because \(T\) is connected and spanning, the complement \(E(G)\setminus E(T)\) cannot contain a bond of \(G\); otherwise deleting that bond would disconnect \(G\) while the tree \(T\) still survives inside the remaining graph. By cycle-cut duality, the complementary dual edges therefore contain no cycle in \(G^*\).
They also have \[ m-(n-1)=f-1=|V(G^*)|-1 \] edges, so an acyclic subgraph of \(G^*\) with exactly \(|V(G^*)|-1\) edges must be a spanning tree.
Planar Separator Theorem
Planar Separator Theorem
There is a constant \(c\) such that every \(n\)-vertex planar graph \(G\) has a vertex set \(S\) with
and every connected component of \(G-S\) contains at most \(2n/3\) vertices.
Why This Is Powerful
- It gives divide-and-conquer algorithms on planar graphs.
- It is one reason many NP-hard problems become subexponential or fixed-parameter tractable on planar inputs.
- It is the structural opposite of expander behavior: planar graphs cannot stay globally tangled at every scale.
Proof Sketch (Lipton-Tarjan Method)
- Triangulate first. Add edges until the graph becomes a triangulation \(H\). Any separator found in \(H\) is also a separator in the original graph \(G\).
- Take BFS layers. Root a BFS tree \(T\) at a vertex \(r\), and let \(L_0, L_1, \dots, L_h\) be the level sets.
- Find a balanced fundamental cycle. In a triangulation, one can choose a non-tree edge \(xy\) so that the simple cycle formed by the tree path \(rTx\), the tree path \(rTy\), and the edge \(xy\) leaves at most \(2n/3\) vertices on each side. This is the geometric heart of the proof.
- Shorten the separator using light layers. Let \(\ell=\lceil \sqrt{n} \rceil\). Among any \(\ell\) consecutive BFS levels, some level has at most \(n/\ell\) vertices. Choose one light level just inside the cycle and one just outside it. Replace long arcs of the cycle by these light layers.
- Count. The two light layers contribute at most \(2n/\ell\) vertices, and the middle part contributes \(O(\ell)\) vertices. Thus \(|S| = O(\ell + n/\ell) = O(\sqrt{n})\). The balance survives, so every remaining component has at most \(2n/3\) vertices.
What to Remember From the Proof
The theorem is not proved by a single counting identity. It combines planar geometry, BFS layering, and a balancing argument. The full technical proof is longer than fits on one board, but the mechanism is worth remembering:
Coloring Planar Graphs
Coloring Language
A proper \(k\)-coloring assigns one of \(k\) colors to each vertex so that adjacent vertices receive different colors. The least possible \(k\) is the chromatic number \(\chi(G)\).
5-Color Theorem
Every planar graph is \(5\)-colorable.
Proof
We use induction on \(n=|V(G)|\). By the Euler corollary, \(G\) has a vertex \(v\) of degree at most \(5\). By induction, \(G-v\) has a proper \(5\)-coloring.
If the neighbors of \(v\) use fewer than five colors, give \(v\) a missing color and we are done. So only one case remains: \(\deg(v)=5\) and its five neighbors \(v_1,v_2,v_3,v_4,v_5\) appear around \(v\) in the plane with five distinct colors \(1,2,3,4,5\).
Look at the subgraph induced by colors \(1\) and \(3\). If \(v_1\) and \(v_3\) lie in different connected components of that subgraph, swap colors \(1\) and \(3\) on the component containing \(v_1\). Then color \(1\) disappears from \(v_1\)'s side, so color \(1\) becomes available for \(v\).
So suppose \(v_1\) and \(v_3\) are connected by a \(1\)-\(3\) Kempe chain. Together with the edges \(vv_1\) and \(vv_3\), that chain forms a Jordan curve. Therefore \(v_2\) and \(v_4\) lie on opposite sides of that curve and cannot be connected by a \(2\)-\(4\) Kempe chain. Swap colors \(2\) and \(4\) on the component containing \(v_2\). Then color \(2\) becomes free for \(v\).
In every case the coloring extends from \(G-v\) to \(G\). Hence every planar graph is \(5\)-colorable.
Why Kempe Chains Matter
Kempe's original proof of the 4-color theorem was wrong, but the chain-switching idea survives perfectly in the proof of the 5-color theorem.
The extra fifth color gives enough room to make the Jordan curve argument work cleanly.
4-Color Theorem
Every planar graph is \(4\)-colorable.
Ideas Behind the 4-Color Theorem
- Assume a smallest counterexample exists.
- Use Euler-type counting and a discharging argument to prove that every planar graph contains at least one configuration from a finite unavoidable list.
- Show that each configuration on the list is reducible, meaning it cannot occur in a minimal counterexample.
- The contradiction shows that no counterexample exists.
The famous 1976 Appel-Haken proof used a large computer check of reducible configurations. Later proofs streamlined the list, and Gonthier gave a fully formalized proof in 2005.
What Changes From Five to Four?
For five colors, the local degree bound \(\deg(v)\le 5\) plus Kempe chains already finishes the argument. For four colors, the same naive Kempe-chain strategy can fail. That is why the proof needs a global reducibility-and-discharging framework instead of a short induction alone.
Straight-Line Embeddings of Planar Graphs
Fary's Theorem
Every simple planar graph admits a plane embedding in which every edge is drawn as a straight-line segment.
Constructive Idea
One route to the theorem is to triangulate the graph and then work with a \(3\)-connected piece. Such graphs admit a canonical ordering of the vertices. Starting from an outer triangle, place the vertices one by one so that each new vertex sees a contiguous block of vertices on the outer face.
Because the placement is controlled geometrically, every new incident edge can be drawn as a straight segment without crossings. At the end, deleting the extra triangulating edges leaves a straight-line drawing of the original graph.
The full proof is technical, but the theorem says something beautifully strong: crossings are never needed merely because we insist on straight edges.
Algorithmic Versions
- Tutte's barycentric method gives convex straight-line drawings for \(3\)-connected planar graphs.
- Schnyder's theory yields grid embeddings of planar graphs on an \((n-2)\times(n-2)\) grid.
- Modern planarity algorithms usually output a combinatorial embedding first, then convert it into straight-line coordinates.
Why This Matters
Fary's theorem turns an abstract topological embedding problem into a geometric drawing problem. It is one of the reasons planar graph visualization can be algorithmic rather than purely hand-crafted.
How to Test Whether a Graph Is Planar
Fast Necessary Checks
- If a simple graph with \(n \ge 3\) has more than \(3n-6\) edges, it is not planar.
- If a simple bipartite graph with \(n \ge 3\) has more than \(2n-4\) edges, it is not planar.
- If you can exhibit a \(K_5\) or \(K_{3,3}\) minor, then the graph is not planar.
These tests are useful, but they are not sufficient. The Petersen graph, for example, satisfies \(m \le 3n-6\) and is still non-planar.
Exact Planarity Testing
Planarity can be tested in linear time. Classical algorithms go back to Hopcroft-Tarjan, and a practical modern implementation is the Boyer-Myrvold algorithm.
What the Algorithms Maintain
- Run a depth-first search and compute lowpoint information.
- Incrementally assemble a rotation system, meaning a cyclic order of edges around each vertex that would define a plane embedding.
- Check whether the partial embedding constraints stay compatible as back edges are inserted.
- If yes, output an embedding. If not, extract a Kuratowski subdivision as a certificate of non-planarity.
A Practical Testing Workflow
| Stage | Tool | Typical Output |
|---|---|---|
| Quick rejection | Euler edge bounds | Immediate non-planarity in dense cases |
| Structural diagnosis | Minor or subdivision search | \(K_5\) or \(K_{3,3}\) witness |
| Definitive test | Linear-time embedding algorithm | Embedding or Kuratowski certificate |
| Drawing stage | Straight-line embedding routine | Crossing-free coordinates |
Takeaways
Main Results
- Euler's formula \(n-m+f=2\) is the source of the standard sparsity bounds for planar graphs.
- Wagner's theorem characterizes planar graphs by the forbidden minors \(K_5\) and \(K_{3,3}\).
- Duality turns faces into vertices and cycles into cuts.
- Planar graphs have balanced separators of order \(O(\sqrt{n})\).
- Every planar graph is \(5\)-colorable, and in fact every planar graph is \(4\)-colorable.
- Every simple planar graph has a straight-line embedding.
- Planarity can be tested in linear time and certified either by an embedding or by a Kuratowski witness.
Proof Patterns to Remember
- Count edges and faces globally, then force a low-degree local configuration.
- Translate a topological obstruction into a forbidden minor or subdivision.
- Use the embedding itself: inside versus outside, Jordan curves, and dual edges crossing primal edges.
- When a proof becomes too global for induction alone, look for separators or discharging.