Chapter 1 Fundamental Concepts Introduction to Graph Theory Douglas B. West July 11, 2002

1. Definitions & Examples 1. Definitions 1. Definitions 2. Graphs as models 2. Graphs as models 3. Matrices and isomorphism 3. Matrices and isomorphism

1.1 Definitions A simple graph G with n vertices and m edges consists of a vertex set V and an edge set E, where each edge is an unordered pair of vertices. uv  E  u and v are adjacent  u  v  u is adjacent to v. No loop and No multi-edge A simple directed graph or digraph G (each edge is an ordered pair of vertices) uv  E  u  v  there is an edge from u to v. u v

1.2 Graphs as models --- PART1 subgraph: a subgraph H of G that V(H)  V(G) and E(H)  E(G). induced subgraph H of G that E(H)=E V(H) (G). complement of G(G): u and v are adjacent in G iff u and v aren ’ t adjacent in G. __ complete graph (clique) : every pair of vertices form an edge. independent set S in G : S  V(G) and G[S] has no edges. ( E G (S)=  ) bipartite graph : if its vertex set can be divided into two independent sets. complete bipartite

1.2 Graphs as models --- PART2 k-partite 5-partite schedule & color path : an ordered list of distinct vertices v 1, …,v k that v i-1 v i is an edge, i=2~k. cycle : all similar to the above and v k v 1 is an edge. => v 1 v k -path connected : a graph G is connected if it has a uv-path for each pair u,v  V(G).

1.3 Matrices (and isomorphism) A(G) adjacency matrix of G : the entry a ij is the number of the edges of v i v j. M(G) incidence matrix of G : row by V & column by E, the entry m ij =1 if v i  e j, and m ij =0 otherwise. In digraph, m ij =+1 if v i : tail of e j, m ij =-1 if v i : head of e j. w x y z G a b c z w x y H a b c non-loop => degree. (symmetric) v  e means v and e are incident. w x y z w x y z A(G) w x y z w x y z A(H) M(G) a b c w x y z M(H) a b c w x y z

1.3 (Matrices and) isomorphism An isomorphism from G to H is a bijection (1-1 & onto) f : V(G)  V(H) that uv  E(G) if and only if f(u)f(v)  E(H). G is isomorphic to H, G  H. G and H are isomorphic if and only if a permutation to rows (columns) of A(G) to obtain A(H). G w x y z c a d b H w x y z w x y z A(G) a b c d a b c d A(H) w y z x a b c d

2. Paths & Proofs 1. Definitions 1. Definitions 2. Connected graphs 2. Connected graphs 3. Contradiction and bipartite graphs 3. Contradiction and bipartite graphs

2.1 Definitions walk : a walk of length k is a sequence v 0,e 1,v 1,e 2, …,e k,v k, walk : a walk of length k is a sequence v 0,e 1,v 1,e 2, …,e k,v k, of vertices and edges such that e i =v i-1 v i for i=1~k. of vertices and edges such that e i =v i-1 v i for i=1~k. trail : a trail is a walk with no repeated edge. trail : a trail is a walk with no repeated edge. path : a path is a walk with no repeated vertex. path : a path is a walk with no repeated vertex.

2.2 Connected graphs component : the maximal connected subgraph. (nontrivial : contains an edge) cut-edge (cut-vertex): an edge (a vertex) whose deletion increases the number of components. cut-edge belongs to no cycle !!! disjoint union (sum) : G+H where V(G)  V(H)= . union : G  H that V(G)  V(H) and E(G)  E(H). join (G  H) : G+H and add the edges {uv | u  V(G),v  V(H)} G H

2.3 Contradiction and bipartite graphs If v is a cut-vertex of a simple graph G, then v is not a cut-vertex of G. _ A graph is bipartite if and only if it has no odd cycle. G-v : S, T G-v : S ’, T ’ _ S’S’ T’T’ S T TT’TT’ S’TS’TSS’SS’ ST’ST’ (=>) trivial (<=) w.l.o.g connected ………….. v N(v) X Y even odd indep.

3. Vertex Degrees & Counting 1. Definitions 1. Definitions 2. Counting 2. Counting 3. The pigeonhole principle 3. The pigeonhole principle

3.1 Definitions degree of v : number of non-loop edges containing v plus degree of v : number of non-loop edges containing v plus twice the number of loops containing v. twice the number of loops containing v.  (G) : maximum degree of G.  (G) : maximum degree of G.  (G) : minimum degree of G.  (G) : minimum degree of G. k-regular :  (G) =  (G) = k. k-regular :  (G) =  (G) = k. isolated vertex : degree=0. isolated vertex : degree=0. Neighborhood : N G (v), N G [v] Neighborhood : N G (v), N G [v] n(G) : order of G, is the number of vertices in G. n(G) : order of G, is the number of vertices in G. e(G) : the number of edges in G. e(G) : the number of edges in G.

3.2 Counting (Degree Sum Formula) If G is a graph with vertex degree d 1, …,d n, then the summation of all d i = 2e(G). By intuitive way or incidence matrix !!! For n  1, there are 2 simple graphs with vertex set {v1, …,vn} such that every vertex degree is even. all simple graphs of size n-1. the number of vertices of odd degree is even.

3.3 The pigeonhole principle (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 Theorem2: If G is a simple graph of n vertices with  (G)  (n-1)/2, then G is connected. u is adjacent to v or not, there are n-1 edges connecting u,v since  (G)  (n-1)/2 n-2 other vertices => common neighbor

* Induction trap Every 3-regular simple connected graph has no cut-edge. 2k vertices smallest : k 4 expansiion => 2(k+1) vertices Any n horses are of the same color. k k 111

4. Degrees & Algorithmic Proof 1. Algorithmic or constructive proof 1. Algorithmic or constructive proof 2. Graphic sequences 2. Graphic sequences 3. Degrees and digraphs 3. Degrees and digraphs

4.1 Algorithmic or constructive proof Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges. degree sequence : the list of vertex degrees, in nonincreasing order, d 1  …  d n. X Y v

4.2 Graphic sequences The nonnegative integers d 1, …,d n are the vertex degrees of some multigraph if and only if the  d i is even. 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. (d 1 =0) d 2 -1, …,d  (=>) trivial. (<=) the number of vertices of odd degrees is even.  d ’ = (  d) - 2 

4.3 Degrees and digraphs out-degree : d + (v) v is tail. (out-neighborhood N + (v)) in-degree : d - (v) v is head. (in-neighborhood N - (v)) v tournament : complete graph and each edge with orientation. Every tournament has a king. (a king is a vertex from which every other vertex is reachable by a path of length at most 2.) x y x is not a king N + (x)  N + (y) finite degree