Introduction to-graph-theory-1204617648178088-2

Graph Theory

A.Benedict Balbuena

Institute of Mathematics, University of the Philippines in Diliman

8.2.2008

A.B.C.Balbuena (UP-Math) Graph Theory 8.2.2008 1 / 47

Definitions

Simple Graph

DefinitionA simple graph G(V , E) consists of a nonempty set V of verticesand a set E of edges such that each edge e ∈ E is associatedwith an unordered pair of distinct vertices, called its endpoints.

Vertices are adjacent if there is an edge connecting the twovertices.An edge is incident to its endpoints.


Definitions

Simple Graph

DefinitionA simple graph G(V , E) consists of a nonempty set V of verticesand a set E of edges such that each edge e ∈ E is associatedwith an unordered pair of distinct vertices, called its endpoints.

Vertices are adjacent if there is an edge connecting the twovertices.An edge is incident to its endpoints.


Definitions

Degree

DefinitionThe degree of a vertex in a simple graph, denoted deg(v), is thenumber of edges incident on it.

degree also means number of adjacent vertices


Definitions

Handshaking Lemma

TheoremGiven a simple graph G(V , E),

2|E | =∑v∈V

deg(v)

Proof.Each edge contributes 1 each to the degree of its endpoints.Thus each edge adds 2 to the total degree of G.


Definitions

TheoremIn any simple graph, there are an even number of vertices ofodd degree

Proof.Let G(V , G) be a simple graph. The sum of all the degrees ofthe vertices is 2|E |, an even number. Let deven be the sum ofdegrees of vertices with even degree and dodd the sum ofdegrees of vertices with odd degree. The total deven is even, thusdodd is even. This implies that there must be an even number ofthe odd degrees. Hence, there must be an even number ofvertices with odd degree.


Paths

Paths and Cycles

DefinitionA path is a sequence of vertices v0, v1, v2, ..., vn where there isan edge connecting vi and vi+1 for i = 0, 1, .., n − 1. A path withthe same starting and endpoint is called a cycle (circuit). Agraph is simple if no edge is repeated.

acyclic graph has no cycles


Paths

Adjacency Matrix

DefinitionThe adjacency matrix of a simple graph G with n vertices is abinary n× n matrix such that each entry ai,j is either zero or one.ai,j = 0 if there is no edge from vi to vj , otherwise it is 1.


Paths

TheoremLet G be a graph with adjacency matrix A with respect to theordering v1, v2, ..., vn and r a positive integer. The number ofdifferent paths of length r from vi to vj is the (i , j)-th entry of Ar


Paths

Connectedness

DefinitionA simple graph is called connected if there is a path betweenevery pair of distinct vertices. A graph that is not connected issaid to be disconnected


Paths

TheoremThere is a simple path between every pair of distinct vertices ofa connected simple graph

Proof.Let u and v be two distinct vertices of a connected simple graphG. SInce G is connected then there is a path from u to v , sayu = x0, x1, ..., xn−1, xn = v . If this were not simple, then there is asubpath where xi = xj for some i and j with 0 ≤ i ≤ j . Meaningthere is a path from u to v of shorter length with vertexsequence u, x1, ..., xi−1, xj , ..., v obtained by deleting the edges ofthe subpath.


Paths

Euler Paths

DefinitionA simple path that contains all edges of a graph is called anEuler path. If this path is also a circuit, it is called an Eulercircuit.

eulerian graphs have euler circuits


Paths

TheoremIf a simple graph has an Euler circuit then every vertex of thegraph has even degree.

Proof.Let G be a graph with an Euler circuit. Start at some vertex onthe circuit and follow the circuit from vertex to vertex, erasingeach edge as you go along it. When you go through a vertexyou erase one edge going in and one edge going out, reducingthe degree of the vertex by 2. Eventually every edge getserased and all the vertices have degree 0. So all vertices musthave had even degree to begin with.

contrapositive??


Paths

CorollaryIf a graph has a vertex with odd degree then the graph can nothave an Euler circuit.

TheoremIf all the vertices of a connected graph have even degree, thenthe graph has an Euler circuit.


Paths

TheoremA connected graph has an Euler path but not an Euler circuit ifand only if it has exactly two vertices of odd degree.

Proof.


Paths

Fleury’s Algorithm

constructs an Euler circuit in a graph (if it’s possible).âreduced graphâ = original graph minus the darkened (alreadyused) edges

1 Pick any vertex to start2 From that vertex pick an edge to traverse, considering

following rule: never cross a bridge of the reduced graphunless there is no other choice

3 Darken that edge, as a reminder that you can’t traverse itagain

4 Travel that edge, coming to the next vertex5 Repeat 2-4 until all edges have been traversed, and you are

back at the starting vertex


Paths

Hamilton Paths

DefinitionA path is called a Hamiltonian path if it visits every vertex of thegraph exactly once. A circuit that visits every vertex exactly onceexcept for the last vertex which duplicates the first one is calleda Hamiltonian circuit.


Paths

TheoremIf G is a connected simple graph with n vertices where n ≥ 3,then G has a Hamilton circuit if the degree of each vertex is atleast

n2

.


Paths

Gray Code

A gray code


Special Graphs

Complete Graph Kn

DefinitionA complete graph on n vertices (clique), denoted by Kn, is thesimple graph where each vertex is adjacent to all the othervertices.

Find a formula for the number of edges in Kn


Special Graphs

Paths and Cycles

DefinitionA path on n vertices Pn, is the simple graph consisting of a path.The end-vertices have degree 1 and all vertices in betweenhave degree 2.

DefinitionA Cycle on n vertices Cn, is the simple graph consisting of acycle where each vertex has degree two.


Special Graphs

Wheels and Stars

DefinitionA wheel on n vertices Wn, is the simple graph consisting of acycle Cn and another vertex which is adjacent to all vertices inthe cycle.

DefinitionA star on n vertices is the simple graph consisting n + 1 vertice,n of which are connected to the n + 1-th vertex.


Special Graphs

Bipartite Graphs

DefinitionA bipartite graph is a simple graph in which the vertices can bepartitioned into two disjoint sets V1 and V2 where every edge isincident to one vertex in V1 and one vertex in V2.


Special Graphs

Complete Bipartite Graph

DefinitionA complete bipartite graph Km,n is the graph that has its vertexset partitioned into two disjoint subsets of m and n vertices,respectively. Moreover, there is an edge between two vertices ifand only if one vertex is in the first set and the other vertex is inthe second set.

Each vertex is connected to all the other vertices in the othersubset.


Special Graphs

N-cube

DefinitionThe n-cube (hypercube) Qn is the graph whose vertices arebinary strings of length n. Two vertices are adjacent if and only ifthe bit strings differ in eactly one position.


Special Graphs

Regular Graphs

DefinitionIf each vertex of a simple graph has degree k then the graph iscalled a regular graph of degree k


Relation Between Graphs

DefinitionA graph H(VH , EH) is a subgraph of G(V , E) if and only ifVH ⊆ V and EH ⊆ E


Relation Between Graphs

Isomorphic Graphs

DefinitionTwo simple graphs G1(V1, E1) and G2(V2, E2) are isomorphic,denoted G1

∼= G2, if there is a one-to-one onto functionf : V1 −→ V2 such that if v1, v2 are adjacent vertices in G1 thenf (v1), f (v2) are adjacent vertices in G2

Edges are preservedThe number of vertices, the number of edges, degrees of thevertices, cycles, longest paths are all invariants underisomorphism.If any of these quantities differ in two graphs, they are notisomorphic.


Planar Graphs

Planar Graphs

DefinitionA graph is planar if there is a way to draw it in the plane withoutedges crossing.

such drawing is called a planar representation, which splits theplane into regionsplanar graphs are relatively easy for humans to grasp sincethere are no crisscrossing edges. Sometimes the advantages ofplanarity are more concrete; for example, when wires arearranged on a surface, like a circuit board or microchip


Planar Graphs

Jordan Curve Theorem

TheoremEvery simple, closed curve separates the plane into two regions,an inside and an outside.


Planar Graphs

Drawing Planar Representations

For connected graphs

Start with a vertex and record faces while drawing.Faces are enclosed by circuitsPrecise rules defining the cycles that are the face boundaries ofa Planar Drawing:

1 Attach edge from vertex on a face to a new vertex.2 Attach edge between nonadjacent vertices on a face.

Every connected planar drawing is obtained by starting with asingle vertex, and repeatedly applying the Rules. Properties ofplanar drawings like Eulerâs formula can be proved by inductionon the number of rule applications used to create a drawing.


Planar Graphs

Euler’s FormulaTheoremIf a connected planar representation of a graph has v vertices, eedges, and f faces, then

v − e + f = 2

Proof.Show this by induction on the number of edges. For the base case, ifthere is no edge then we have 1 verte and 1 face which satisfies theequation. We assume that given n number of edges, v − e + f = 2. Ifwe add an edge, we let it attach it to a new vertex or to an old vertex.For the first one, we choose a face and add an edge with a newvertex, thus v + 1− e − 1 + f = 2. On the other hand, we choose aface and add an edge across it. This splits the face into two new faces,thus v − e − 1 + f + 1 = 2.


Trees

DefinitionA simple graph is called a tree if each pair of distinct verticeshas exactly one path.

is a tree connected? do trees have cycles?


Trees

TheoremAny tree with more than one vertex has at least one vertex ofdegree 1.

Proof.Let v0 and vn be two distinct vertices. Then there is a pathconnecting v0 to vn. By the definition of a tree, there is only oneedge incident on either v0 or vn. Hence, either deg(v0) = 1 ordeg(v1) = 1.


Trees

TheoremA tree with n vertices has exactly n − 1 edges.

Proof.We show this by induction on n.Let P(n) = Any tree with n vertices has n − 1 edges.P(1) is valid since a tree with one vertex has zero edges.Suppose that P(n) holds, we show P(n + 1) holds, any tree withn + 1 vertices has n edges.Indeed, let T be any tree with n + 1 vertices. Since n + 1 ≥ 2then by the previous theorem, T has a vertex v of degree 1. LetT0 be the graph obtained by removing v and the edge attachedto v . Then T0 is a tree with n vertices. By the inductionhypothesis, T0 has n − 1 edges and so T has n edges.

is the converse true??A.B.C.Balbuena (UP-Math) Graph Theory 8.2.2008 34 / 47

Trees

TheoremAny connected graph with n vertices and n − 1 edges is a tree.

A rooted tree is a tree in which a particular vertex is designatedas the root.The level of a vetex v is the length of the simple path from theroot to v .The height of a rooted tree is the maximum length of a path inthe tree.


Trees

DefinitionLet T be a rooted tree with root v0. Suppose v0, v1, ..., vn is asimple path in T and x , y , z are three vertices. Then

1 vn−1 is the parent of vn

2 v0, v1, ..., vn−1 are the ancestors of vn:3 vn is the child of vn−1

4 If x is an ancestor of y then y is a descendant of x5 If x and y are children of z then x and y are siblings6 If x has no children, then x is a leaf.7 The subtree of T rooted at x is the graph whose vertices

consist of x and its descendants and whose edges areedges in the graph T with e


Trees

DefinitionA binary tree is a rooted tree such that each vertex has at mosttwo children, left child or a right child. A full binary tree is abinary tree in which each vertex has either two or zero children.

maximum degree in a binary tree??


Trees

DefinitionT is a spanning tree of a graph G if it is a subgraph of G that is atree and contains all of the vertices of G.


Coloring

Coloring

DefinitionA coloring of a simple graph is the assignment of a color to eachvertex so that no two adjacent vertices have the same color

DefinitionA graph is k − colorable if each vertex can be assigned one ofthe k colors so that adjacent vertex get different colors.


Coloring

TheoremA graph with maximum degree at most k is (k+1)-colorable.

DefinitionThe chromatic number of a graph is the least number of colorsneeded to color a graph


Digraphs

Digraphs

DefinitionA directed graph or digraph D(X , A) consists of a nonempty setV of vertices and a set E of arcs such that each arc a ∈ A isassociated with an ordered pair of distinct vertices.


Digraphs

Digraphs

DefinitionA directed graph or digraph D(X , A) consists of a nonempty setV of vertices and a set E of arcs such that each arc a ∈ A isassociated with an ordered pair of distinct vertices.


Digraphs

In and Out Degrees

DefinitionWhen (u, v) is an arc a digraph then u is called the initial vertexand v is called the terminal vertex. In a directed graph, thein-degree of a vertex v , denoted by deg−(v), is the number ofedges with v as their terminal vertex. Similarly, the out-degreeof v , denoted by deg+(v), is the number of edges with v as aninitial vertex.

denote arc (u, v) by u → vu → v means u in-neighbor of v and v out-neighbor of uNote that deg(v) = deg+(v) + deg−(v)


Digraphs

DefinitionThe set of out-neighbors of a vertex v and the vertex v itself iscalled the vicinity of v , denoted N(v).

DefinitionA vertex is a source if it has no in-neighbor and is an in-neighborof at least one vertex. A vertex is a sink if it has no outneighborand is an out-neighbor of at least one other vertex


Digraphs

TheoremGiven digraph D(X , A),

|E | =∑v∈X

deg+(v) =∑v∈X

deg−(v)


Digraphs

Directed Paths

DefinitionA directed path of length n (where n is a positive integer) from uto v in a digraph is a sequence of points u = v0, v1, v2, ..., vn = vwhere there is an arc connecting vi and vi+1 for i = 0, 1, .., n − 1.A path with the same starting and endpoint is called a cycle(circuit). A directed path is simple if no edge is repeated.


Digraphs

Connectedness

DefinitionA digraph is strongly connected if there is a path from a to b andfrom b to a whenever a and b are vertices in the graph.

DefinitionA digraph is weakly connected if there is a path between anytwo vertices in its underlying undirected graph.


Digraphs

Transitive Digraphs

DefinitionA transitive digraph is a digraph D(V , A) such that for any threevertices u, v , x ∈ V , if u → v and v → x then u → x .

DefinitionThe underlying undirected graph of a transitive digraph is calleda comparability graph.


Education

Introduction to-graph-theory-1204617648178088-2