1. Summing degree sequences work out degree sequence, and sum

2. The handshaking lemma weak version: The sum of degrees of a graph G is even strong version: The sum of degrees of a graph G is double the number of edges why? proof?

3. Proof by induction on the number of edges... If G has 3 edges, then If G has 2 edges, then If G has 1 edges, then If G has 0 edges, then

4. Proof by induction on the number of edges... If G has 3 edges, then If G has 2 edges, then If G has 1 edges, then If G has 0 edges, then

5. Proof by induction begin the induction: If G has 0 edges, then Proof: If G has no edges, then all the degrees of its vertices must be zero. The sum of the degrees is therefore = 2 E(G)

6. Proof by induction rough thinking for next step: If for graphs with no edges then for graphs with one edge. Step: If G has one edge, then let G’ be G with the edge removed. G’ has no edges, so Replace the edge - two vertices increase degree by 1, so sum increases by 2.

7. Proof by induction rough thinking for next step: If for graphs with one edges then for graphs with two edges. Step: If G has two edges, then let G’ be G with an edge removed. G’ has one edge, so Replace the edge - two vertices increase degree by 1, so sum increases by 2.

8. Proof by induction general step: If for graphs with k edges then for graphs with k+1 edges. Inductive proof: If G has k+1 edges, then let G’ be G with an edge removed. G’ has k edges, so Replace the edge - two vertices increase degree by 1, so sum increases by 2.

9. Degree sequences when is there a graph with a given degree sequence?

10. Degree sequences Given a degree sequence - check that the sum is even (if not, quote the handshaking lemma) - apply a rule, iteratively: -> -> -> -> stop if this process ends up with, then a graph is drawable with the given degree sequence. why?

11. Graphs Properties which remain the same under isomorphism: number of nodes number of edges connectedness degree sequence if all of these are equal, the graphs may or may not be isomorphic

12. Graphs More properties : Eulerian Hamiltonian plane planar NB: “Euler” is pronounced “oiler”

13. Eulerian? try to visit all edges exactly once using a cycle

14. Eulerian definition A graph G is Eulerian if and only if there is a cycle which includes all edges once (called an “Euler cycle”)

15. Eulerian theorem A graph G is Eulerian if and only if all vertices have even degree.

16. Hamiltonian? try to visit all vertices exactly once using a cycle

17. Hamiltonian definition A graph G is Hamiltonian if and only if there is a cycle which visits all vertices once (called a “Hamiltonian cycle”) no known theorem!

18. Plane? A graph is plane if there are no edge-crossings in the drawing

19. Plane vs. Planar plane not planenot plane planar planarnot planar In general - a graph is planar if it’s isomorphic to a plane graph. So all plane graphs are planar. And all non-planar graphs are not plane.

20. Complete graphs K n have n vertices and all possible edges (how many edges?) K r,s have r + s vertices, split into two sets, and all possible edges between the r-set and the s-set. (how many edges?)

21. Complete graphs Which of K n and/or K r,s are Eulerian/Hamiltonian/planar Answer for K 1 K 2 K 3 K 4 K 5 and forK 1,1 K 1,2 K 2,2 K 2,3 K 3,3 Start by drawing them.