Matthew Bowling Euler’s Theorem Mathfest Spring ‘15

Definition: A graph or general graph is an ordered Triple G = (V,E,Φ), where: 1. V≠Ø 2. V∩E=Ø 3. Φ: E-> P(V) is a map such that |Φ(e)| ∈ {1,2} for each e ∈ E

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(e 1 )={1,2} says that edge 1 in the graph connects vertex 1 and vertex 2.

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

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

V-E+f=2? V=4 E=5 F= =2 V-E+f=2? V=5 E=5 F= =2 V-E+f=2? V=3 E=2 F= =2

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

IF T is a simple graph on n vertices, then the following statements are equivalent: 1. T is a tree 2. T has v-1 edges and no cycles 3. T has v-1 edges and is connected 4. T is connected and each edge is a bridge 5. Between every pair of distinct vertices in T there is exactly one path 6. T has no cycles, but add an edge to T between a pair of nonadjacent vertices and exactly one simple cycle is formed.

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.

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

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.

Applying the induction hypothesis to G’=G-e we have: V-E’+F’=2 V-(E-1)+(F-1)=2 V-E+F+(1-1)=2 V-E+F=2 Thus, our Formula is proven for any F≥1.

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.

Euler’s Formula does not work in higher genus graphs However, 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)

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 T g. 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.

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-1 If 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).

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.

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.

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.