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Outline
Discrete Mathematics
S Benecke
Applied Mathematics DivisionDepartment of Mathematical Sciences
University of Stellenbosch, South Africa
Hons Program PresentationOctober 10, 2011
S Benecke Discrete Mathematics
Outline
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Graph Theory
Some of the TopicsTrees and Searching AlgorithmsVertex Traversal and HamiltonianGraphsEdge Traversl and EulerianGraphsPlanar GraphsScheduling and Graph ColouringTournaments
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Heuristic Methods in Graph Theory
Classical ProblemsThe Travelling Salesman ProblemThe Chinese Postman Problem
Selected TopicsGenetic AlgorithmsGreedy AlgorithmsSimulated AnnealingTabu SearchGenetic Algorithms
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Cryptology
Encryption/DecryptionHow do I encrypt a message so that only the receiver isable to understand it?How do I decipher a message that I intercept?Application of Number Theory
ContentBlock CiphersStream CiphersPublic Key Systems
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Coding Theory
Error CorrectionDesigning of error-correcting codes to ensure the integrityof informationAn application of vector spaces over finite fields
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Outline
1 Graph Theory
2 Heuristic Methods
3 Cryptology
4 Coding Theory
5 Proposed Projects
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Proposed Projects
GroblerThe appearance of Fibonacci numbers in the arrangementof leaves and florets of plantsThe existence and designing of Mutually Orthogonal LatinSquares
BeneckeA survey and development of algorithmic methods forgraph domination problemsAn analysis and solution to the Guess Who Problem
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
The Cartesian Product
DefinitionThe Cartesian product of two graphs G and H,V(G) = {v1, v2, . . . , vm}, V(H) = {w1,w2, . . . ,wn}
Denoted G � H
V(G � H) = {(vi,wj) : i = 1, 2, . . . ,m, j = 1, 2, . . . , n}(vi,wj)(vk,wl) ∈ E(G � H) if and only if
j = l and vivk ∈ E(G), ori = k and wjwl ∈ E(H)
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Example
Consider G ∼= P3 and H ∼= C3
DefinitionVertex set V(G � H) = {(vi,wj) :i = 1, 2, . . . ,m, j = 1, 2, . . . , n}(vi,wj)(vk,wl) ∈ E(G � H) iff
j = l and vivk ∈ E(G), ori = k and wjwl ∈ E(H)
H
v1
v2
v3G
(v1, w1)
(v2, w1)
(v3, w1)
(v1, w2) (v1, w3)
(v2, w3)
(v3, w3)
(v3, w2)
w1 w2
(v2, w2)
w3
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Domination
DefinitionA subset D ⊆ V(G) is a dominating set if any vertex u 6∈ Dis adjacent to some vertex v ∈ D.The domination number γ(G) is the minimum cardinalityover all dominating sets D of G.
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Domination Algorithms
ProjectSurvey best algorithms to determine the dominationnumber of the Cartesian product graphStudy and implement the method by Livingston & Stout, asdescribed by Benecke & MynhardtSurvey best algorithms for and investigate application toother graph products and/or other domination parameters
ContentGraph TheoryProgramming
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
The Guess Who Game
QuestionsGeneralize classic game to k ≥ 1 mystery people.What is the best strategy?What is the best question at any stage?How does one design a balanced game board?Under which conditions can the game always be resolved?
ContentGame TheoryProbability TheoryProgramming
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
References
A MENEZES, P VAN OORSCHOT AND S VANSTONE,Handbook of Applied Cryptography,CRC Press, 1996.
JA BONDY AND USR MURTY,Graduate Texts in Mathematics - Graph Theory,Springer, 2008.
Z MICHALEWICZ AND DB FOGEL,How to Solve it: Modern Heuristics,Springer, 2000.
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Cryptology
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Graph Theory
S Benecke Discrete Mathematics
Graph TheoryHeuristic Methods
CryptologyCoding Theory
Proposed Projects
Heuristic Methods
S Benecke Discrete Mathematics