Chapter 1 Fundamental Concepts II Pao-Lien Lai 1

Definitions Counting The pigeonhole principle Graphic sequences Degrees and digraphs 2

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

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

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).

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Example 21 The list is graphic w v u The realization is not unique!

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)

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

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

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

Application Finite state machine Markov chain 26

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

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        011

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

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

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

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

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)

Proposition In a digraph G, 34

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

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

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

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 ◦ works

Example 39 D4D4

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

Example

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.

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

Exercise 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

Exercise or 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