V-E+F = 2

Matthew BowlingEuler’s TheoremMathfest Spring ‘15

Finally: What is A Graph?

Definition: A graph or general graph is an ordered Triple G = (V,E,Φ), where:

1. V≠Ø2. V∩E=Ø3. Φ: E-> P(V) is a map such that |Φ(e)| {1,2} for ∈each e E∈

What does that Definition Mean?

1. V cannot be empty, so to have a graph, you need vertices such that V≥1.2. The set V and set E do not ever contain the same elements, and so their intersection is always an empty set.3. Φ denotes the relation between which vertices are connected by an edge.

For example, Φ(e1)={1,2} says that edge 1 in the graph connects vertex 1 and vertex 2.

Basic Graph Theory Examples

Basic Graph Theory Examples (cont.)

Basic Graph Theory Examples (cont.)

Basic Graph Theory Examples (cont.)

Getting more advanced: Planar Graphs

Graphs drawn in a plane where intersections of distinct edges occur only at a vertexGraph G can be embedded on a sphere, or drawn onto it without edges crossing.Examples:

SphereTorus

Finally: Euler’s Formula

Theorem: Every connected planar graph with v vertices, e edges, and f faces satisfies the following equation:

F in a planar graph is an enclosed or unbounded region

V-E+F=2

Examples of Euler’s FormulaV-E+f=2?V=4E=5F=34-5+3=2

V-E+f=2?V=5E=5F=25-5+2=2

V-E+f=2?V=3E=2F=13-2+1=2

Examples of Euler’s Formula

V-E+F=2?V=4E=6F=54-6+5=2?

V-E+F=2?V=4E=6F=44-6+5=2?

A Theorem to Note

IF T is a simple graph on n vertices, then the following statements are equivalent:

1. T is a tree2. T has v-1 edges and no cycles3. T has v-1 edges and is connected4. T is connected and each edge is a bridge5. Between every pair of distinct vertices in T there is exactly one path6. T has no cycles, but add an edge to T between a pair of nonadjacent vertices and exactly one simple cycle is formed.

One more Theorem to note

Jordan Curve Theorem: IF G is a plane embedding of a cycle graph then G has precisely two faces. One face is formed by the region “inside” the cycle and one face is formed by the region “outside” the cycle.

Finally: Euler’s Formula Proof

We will prove this inductively with F:Let G be a planar graph that has been drawn in the plane with no edges crossing, with V vertices, E edges, and F faces. For the base case f=1, G is a tree. Then we will have V(T)=V, E(T)=V-1 (By previous theorem of a tree), F(T)=1.So if we fill in the formula V-E+F=2 for this case we have:

V-E+F = V-(V-1)+1=2 =>V-V+1+1=2=>2=2For the case F=1, the formula checks out.Now we will assume the result is true for connected planar graphs with F-1 faces.

Finally: Euler’s Formula Proof (cont.)

For G, we must prove that this formula holds true when F>1. If F>1, then there must exist a cycle within G, and so G is not a tree (by previous theorem).If edge E is in our cycle, then it is on the border of two faces which we will call S, and S’. If the edge is removed, then there will exist a new single face S’’. Removing this edge creates V vertices(which is unaffected), E’=e-1 edges, and F’=f-1 faces.

Finally: Euler’s Formula Proof (cont.)

Applying the induction hypothesis to G’=G-e we have:

V-E’+F’=2V-(E-1)+(F-1)=2V-E+F+(1-1)=2V-E+F=2

Thus, our Formula is proven for any F≥1.

What now?Now, we will discuss higher genus graphs.Genus of a graph: least integer g such that G can be embedded on a torus with g holes.The ‘genus’ of a graph can be viewed as a parameter used to measure how far a graph is from being planar.

Does Euler’s Formula work in higher Genus?

Euler’s Formula does not work in higher genus graphsHowever, There is a formula for an extension of Euler’s formula to work with higher genus graphs.Extension of Euler’s Formula: V-E+F=2-2(g)

Proof of Generalized Euler’s Formula

We will prove this theorem inductively on the Genus g.Let g=0, then this is the standard form for Euler’s Formula, which has already been proven.Let g≥1, and assume that G is a fixed graph of genus g, which is embedded on Tg. We can assume there is a cycle where the handle meets the sphere. We will cut off the handle at the point the handle attaches to the cycle.

Proof of Generalized Euler’s FormulaCont.

By cutting at the cycle, there creates two cycles, one on the sphere, and one on the handle. By filling in the space, we created a new graph G’ which has one less handle on it. This creates g-1 genus for the graph. Therefore g’=g(G’)=g-1If the cycle, C, has i vertices and i edges, then the number of V’ vertices and E’ edges is given by V’=V+i, E’=E+i and F’=F+2 (the 2 comes from a cycle having 2 faces).

Proof of Generalized Euler’s FormulaCont.

By the induction hypothesis we have:V’-E’+F’=2-2g’So: V-E+F= (V’-i)-(E’-i)+(F’-2)=(V’-E’+F’)-2=(2-2g’)-2=2-2(g’+1)=2-2g

Thus V-E+F=2-2(g), proving our formula.

Final Thoughts

Any planar graph, with non-intersecting edges can be figured into Euler’s Formula.For those graphs which require many handles, the generalized Euler’s Formula will work to determine it’s value.Graph Theory has many interesting facts, which work with other subjects in mathematics (ex. Topology) to discover.

Special Thanks

Dr. Roblee For teaching Graph Theory, and putting so much energy into Mathfest for students to have this opportunity.Dr.Belyi For teaching Topology so I have a general understanding of the idea of a Torus and some of the Topological properties in planar graphs.The entire department of Mathematics For being so open and helpful.