Problems of Graph Theory

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Overview

Path Problems

Isomorphic Graphs

Degree Sequences

Shortest Paths

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Outline

Path Problems

Isomorphic Graphs

Degree Sequences

Shortest Paths

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The Konigsberg Bridges

Can one find a route that would cross each of the sevenbridges exactly once?

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Euler’s Approach

The graph on the vertices A,B ,C ,D with edgesa, b, c , d , e, f , g does not have an Euler trail.

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Graph Terminology

I A simple graph has no loops or multiple edges.

I A walk is a sequence of edges, each consecutive pair ofwhich is incident to a common vertex.

I A closed walk is one in which the initial and final verticescoincide.

I A trail is a walk in which all edges are distinct.

I A circuit is a closed trail.

I A path is a trail in which all vertices distinct (with thepossible exception of the initial and final verticescoinciding).

I A cycle is a closed path.

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Euler Trails

Determine whether the graph has an Euler trail (i.e., a trailwhich contains every edge of the graph).

b c d

a

g

e f

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Hamilton PathsDetermine whether the graph has a Hamilton path (i.e., apath which contains every vertex of the graph).

a b

c

d

e

f

g

h

i

j k

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Hamilton Cycles

Determine whether each graph has a Hamilton cycle (i.e., acycle which contains every vertex of the graph).

a bc

d

e

f

g

h

ij k

a

b

c

d

e

f

g

h

i

j

k

l

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Outline

Path Problems

Isomorphic Graphs

Degree Sequences

Shortest Paths

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Isomorphic Graphs

DefinitionTwo graphs are isomorphic if they are essentially the same,even though they may appear different, in the sense that onecan be transformed into the other by moving vertices aroundand changing the lengths of some of the edges that join them.

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Example

Determine whether the following graphs are isomorphic.

a

b

d

c

a

b

c

d

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Example

Determine whether the following graphs are isomorphic.

a

b

c

d

f e

a

b

c

d

f e

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Example

Determine whether the following graphs are isomorphic.

a

b

c

d

e

a

b

c

d

e

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Outline

Path Problems

Isomorphic Graphs

Degree Sequences

Shortest Paths

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Definition of Degree Sequence

DefinitionThe degree sequence of a simple graph is a list, or sequence,of all of the degrees of the vertices in the graph including anyrepetitions.

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Example

The following graph has degree sequence (5, 5, 4, 4, 4, 4, 4).

b c d

a

g

e f

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Question

Which sequences are degree sequences of graphs?

Note that there is no simple graph having degree sequence(3, 2, 2). Why?

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Question

Which sequences are degree sequences of graphs?

Note that there is no simple graph having degree sequence(3, 2, 2). Why?

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Handshaking Lemma

LemmaThe sum of the degrees of the vertices of a graph is twice thenumber of edges.

CorollaryIn any graph, the sum of the vertex degrees is even.

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Example

Determine which of the following sequences are degreesequences for some simple graph.

(a) (5, 5, 5, 4, 4, 3, 3)

(b) (7, 6, 5, 4, 3, 2, 1)

(c) (6, 5, 4, 3, 2, 2, 1)

(d) (6, 5, 4, 3, 2, 1, 0)

(e) (3, 3, 2, 2, 2, 1, 1)

(f) (5, 5, 4, 4, 4, 3, 3)

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Existence of Vertices of Equal Degree

PropositionIn any simple graph having two or more vertices, there mustexist at least two vertices having the same degree.

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Outline

Path Problems

Isomorphic Graphs

Degree Sequences

Shortest Paths

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Shortest Path Problem

Find a shortest route between vertices in a given graph, whichmay be undirected, directed, weighted, etc.

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Undirected Graph

Find a shortest path (i.e., a path with fewest edges)from a to h.

a

b

c

d

e

f

g

h

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Directed Graph

Find a shortest path (i.e., a path with fewest edges) path froma to h, respecting the edge directions.

a

b

c

d

e

f

g

h

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Acknowledgements

Statements of results follow the notation and wording ofAnderson’s First Course in Discrete Mathematics.

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