1. Chapter 1 Fundamental Concepts II Pao-Lien Lai 1

2. Definitions Counting The pigeonhole principle Graphic sequences Degrees and digraphs 2

3. Definitions degree of v : ◦ number of non-loop edges containing v plus twice the number of loops containing v.  (G) : (\Delta) maximum degree of G.  (G) : (\delta) minimum degree of G. k-regular :  (G) =  (G) = k. 3

4. Definitions Isolated vertex : degree=0. Neighborhood : N G (v), N G [v] n(G), |G| : ◦ order of G, is the number of vertices in G. e(G) : the number of edges in G. 4

5. Counting 5 (Degree Sum Formula) If G is a graph with vertex degree d 1,…,d n, then the summation of all d i = 2e(G).

6. Counting In a graph G, the average vertex degree is, and hence 6 Every graph has an even number of vertices of odd degree. No graph of odd order is regular with odd degree. A k-regular graph with n vertices has nk/2 edges.

7. Example k-dimensional cube (hypercube Q k ) Vertices: k-tuples with entries in {0,1} Edges: the pairs of k-tuples that differ in exactly one position. j-dimensional subcube: a subgraph isomorphic to Q j. 7 Q3Q3

8. Example Structure of hypercubes ◦ Parity of vertex: the number of 1s ◦ Two independent sets  Each edge of Q k has an even vertex and an odd vertex.  Bipartite graph ◦ k-regular ◦ n(Q k )=2 k. e(Q k )=k2 k-1. ◦ Two subgraphs of Q 3 are isomorphic to Q 2. 8

9. Counting If k > 0, then a k-regular bipartite graph has the same number of vertices in each partite set. 9

10. The Graph Menagerie 動物園 10 triangle claw 爪 paw 爪子 kite 鳶

11. Petersen graph The simple graph whose Vertices: ◦ 2-element subsets of 5-element set Edges : ◦ the pairs of disjoint 2-element subsets 11

12. Petersen graph 12 vertex-transitive Girth 5 Two nonadjacent vertices  they have exactly one common neighbor Girth of a graph: the length of its shortest cycle

13. The pigeonhole principle 13 (Pigeonhole Principle) If a set consisting of more than kn objects is partitioned into n classes, then some class receives more than k objects. Theorem1: Every simple graph with at least two vertices has two vertices of equal degree. {0,1,……,n-1}0 and n-1 both occurs impossibly

14. The pigeonhole principle 14 Theorem 2: If G is a simple graph of n vertices with  (G)  (n- 1)/2, then G is connected.

15. Example Let G be the n-vertex graph with components isomorphic to and. 15 G is disconnected

16. * Induction trap 16 Every 3-regular simple connected graph has no cut-edge. False conclusion!! Counterexample Cut edge

17. Degree sequence 17 degree sequence : the list of vertex degrees, in nonincreasing order, d 1  …  d n.

18. Proposition The nonnegative integers d 1, d 2, …, d n are the vertex degrees of some graph if and only if is even. 18

19. Graphic sequences 19 graphic sequence : a list of nonnegative numbers that is the degree sequence of some simple graph

20. Example A recursive condition 20 The lists 1,0,1 and 2,2,1,1 are graphic The list 2,0,0 is not graphic

21. Example 21 The list 33333221 is graphic 33333221 w2223221 3222221 v111221 221111 u10111 11110 The realization is not unique!

22. Graphic sequences 22 Graphic Theorem: For n > 1, the nonnegative integer list d of size n is graphic if and only if d’ is graphic, where d’ is the list of size n-1 obtained from d by deleting its largest  and subtracting 1 from its  next largest elements. (The only 1-element graphic sequence is d 1 =0)

23. Digraphs 23 A directed graph or digraph G is a triple consisting of a vertex set V(G), and edge set E(G), and a function assigning each edge an ordered pair of vertices Tail: the first vertex of the ordered pair Head: the second vertex of the ordered pair Endpoints: tail and head An edge: from tail to head tail head

24. Digraphs Loop: an edge whose endpoints are equal Multiple edges: ◦ edges having the same ordered pair of endpoints. Simple graph: ◦ each ordered pair is the head and tail of at most one edge ◦ One loop may be present at each vertex 24

25. Digraphs In a simple graph ◦ An edge uv:  tail u and head v  From u to v ◦ v is a successor of u ◦ u is a predecessor of v 25 u v

26. Application Finite state machine Markov chain 26

27. Digraphs Path ◦ A simple digraph whose vertices can be linearly ordered so that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering Cycle ◦ Defined similarly using an ordering of the vertices on a circuit. 27

28. Example Functional digraph of f ◦ The simple digraph with vertex set A and edge set {(x,f(x):x  A)} ◦ For each x, the single edge with tail x points to the image of x under f. Permutation 28 001  010 010  100 100  001 111  111 011  110 110  101 101  011

29. Digraphs Underlying graph 相關圖 of a digraph D ◦ The graph G obtained by treating the edges of D as unordered pairs ◦ The vertex set and edge set remain the same ◦ The endpoints of an edge are the same in G as in D ◦ But the edge become an unordered pair in G. 29

30. Example 30 A(G) M(G) A(D) M(D)

31. Digraphs Weakly connected ◦ Underlying graph is connected Strongly connected (strong) ◦ For each ordered pair u,v of vertices, there is a path from u to v. Strong components ◦ Maximal strong subgraphs 31

32. Example 32 Not strongly connected 5 strong components 1 strong component 3 strong components

33. Degrees and digraphs 33 Out-degree : d + (v) v is tail. (out-neighborhood N + (v) ) In-degree : d - (v) v is head. (in-neighborhood N - (v) ) Minimum in-degree:  - (G) Maximum in-degree:Δ - (G) Minimum out-degree:  + (G) Maximum out-degree: Δ + (G)

34. Proposition In a digraph G, 34

35. Eulerian Digraphs Eulerian trail ◦ A trail containing all edges Eulerian circuit ◦ A closed trail containing all edges Eulerian ◦ A digraph is Eulerian if it has an Eulerian circuit 35

36. Lemma If G is a digraph with  + (G)  1, then G contains a cycle. The same conclusion holds when  - (G)  1. 36 Maximal path P

37. Theorem A digraph is Eulerian if and only if d + (v)=d - (v) for each vertex v and the underlying graph has at most one nontrivial component. 37

38. Application De Bruijn cycles ◦ 2 n binary strings of length n ◦ Is there a cyclic arrangement of 2 n binary digits such that the 2 n strings of n consecutive digits are all distinct? For example: ◦ n=4 ◦ 0000111101100101 works 38 0000 0001 0011 0111 1111 1110 1101 1011 0110 1100 1001 0010 0101 1010 0100 1000

39. Example 39 D4D4

40. Theorem The digraph D n is Eulerian, and the edge labels on the edges in any Eulerian circuit of D n from a cyclic arrangement in which the 2 n consecutive segments of length n are distinct. 40

41. Example 41 0000 0001 0011 0111 1111 1110 1101 1011 0110 1100 1001 0010 0101 1010 0100 1000 0 1 2 3 4 56 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

42. Degrees and digraphs 42 An orientation of graph G: a digraph D obtained from G by choosing an orientation (x  y or y  x) for each edge xy  E(G). An orientation graph is an orientation of a simple graph tournament 比賽 : complete graph and each edge with orientation.

43. Example Consider an n-team league where each team plays every other exactly once. ◦ For each pair u,v  Include the edge uv if u wins  Include the edge vu if v wins At the end ◦ There is an orientation of K n ◦ The score of a team is its outdegree 43

44. Exercise 1.3.8 Which of the following are graphic sequences? Provide a construction or a proof of impossibility for each ◦ (5,5,4,3,2,2,2,1) ◦ (5,5,4,4,2,2,1,1) ◦ (5,5,5,3,2,2,1,1) ◦ (5,5,5,4,2,1,1,1) 44

45. Exercise 1.4.19 or 1.4.20 A digraph is Eulerian if and only if d + (v)=d - (v) for each vertex v and the underlying graph has at most one nontrivial component. 45