1. 02/01/05Tucker, Sec. 1.31 Applied Combinatorics, 4th Ed. Alan Tucker Section 1.3 Edge Counting Prepared by Joshua Schoenly and Kathleen McNamara

2. 02/01/05Tucker, Sec. 1.32 Theorem 1 Statement: In any graph, the sum of the degrees of all vertices is equal to twice the number of edges. where E is the number of edges, d is the degree of any vertex, n d is the number of vertices of degree d, and v is any vertex in the graph.

3. 02/01/05Tucker, Sec. 1.33 Example 1: Use of Theorem 1 Suppose we want to construct a graph with 20 edges and have all vertices of degree four. How many vertices must the graph have? It must have 10 vertices!

4. 02/01/05Tucker, Sec. 1.34 Proof: Summing the degrees of all vertices counts all instances of some edge being incident to some vertex. But, each edge is incident with two vertices, and so the total number of such edge- vertex incidences is simply twice the number of edges.

5. 02/01/05Tucker, Sec. 1.35 Corollary: In any graph, the number of vertices of odd degree is even. Proof of Corollary: Because twice the number of edges must be an even integer, the sum of degrees must be an even integer. For the sum of degrees to be an even integer, there must be an even number of odd integers in the sum.

6. 02/01/05Tucker, Sec. 1.36 Definitions Component- connected pieces of a graph Component 1Component 3Component 2 Length- the number of edges in a path. a b c d e Path abde has length 3. Path eabcd has length 4.

7. 02/01/05Tucker, Sec. 1.37 Definitions Bipartite- a graph G is bipartite if its vertices can be partitioned into two sets, v 1 and v 2, such that every edge joins a vertex in v 1 with a vertex v 2. v1v1 v2v2

8. 02/01/05Tucker, Sec. 1.38 Definitions Complete Bipartite- a bipartite graph in which every vertex on one side is connected to every vertex on the other side. For Example: K 3,4

9. 02/01/05Tucker, Sec. 1.39 Theorem 2 Statement: A graph G is bipartite if and only if every circuit in G has an even length. L=4

10. 02/01/05Tucker, Sec. 1.310 Proof: Bipartite implies even length circuits a b d f e c a f b d c e G

11. 02/01/05Tucker, Sec. 1.311 Proof (part 2): Even length circuits imply bipartite f b d c e G a b d f e c a

12. 02/01/05Tucker, Sec. 1.312 Example 4 A B C D E M T U V W X Y Z Rules: 1.Hikers start at points A and Z 2.Hikers must stay at the same altitude at all times 3.Stops must have at least one hiker on a peak or valley 4.Objective: Hikers must reach M at the same time

13. 02/01/05Tucker, Sec. 1.313 The hikers must be at the same altitude, so when hiker A climbs to point C, hiker Z must climb to point X. Then when hiker A descends to D, hiker Z must then descend to Y in order to be at the same altitude. A B C D E M T U V W X Y Z (A,Z) (C,X) (D,Y)

14. 02/01/05Tucker, Sec. 1.314 Solution to Example 4 (A,Z) (C,X) (D,Y)(D,U) (E,W) (B,U) (M,M) (C,V)(C,T) A B C D E M T U V W X Y Z

15. 02/01/05Tucker, Sec. 1.315 Class Problem – 1.3 / 3 What is the largest possible number of vertices in a graph with 19 edges and all vertices of degree at least 3? (Hint – use Theorem 1)

16. 02/01/05Tucker, Sec. 1.316 Solution to Problem First try with only degree 3: But 38 isn’t divisible by 3, so that doesn’t work. Next try with all vertices having degree 3, except one with degree 4: But 34 isn’t divisible by 3, so that doesn’t work either. Next try with all vertices having degree 3, except two with degree 4: That works! So the largest number of vertices is 12.