13Hamiltonian_10_v2

# 13Hamiltonian_10_v2 - ENGG1007 Foundations of Computer...

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1 ENGG1007 Foundations of Computer Science Graphs Graphs Hamiltonian Graphs Hamiltonian Graphs Professor Francis Chin, Dr SM Yiu November 11/12, 2010 Chapter 9.5

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2 ENGG1007 FCS Word Puzzle Word Puzzle Given these words: CAR, CUB, DIM, HEN, HUT, MOB, RED, SAW, SON, WIT. The task is to arrange these words in a closed chain such that any two adjacent words must share a common letter. The right list is correct except RED and SON. This can be reduced to a graph problem. V = set of words Two words are connected by an edge if they share a common letter. This results the Patersen Graph. The problem is to find a circuit which visits every vertex exactly once. (Hamiltonian Circuit)
3 ENGG1007 FCS Patersen Graph Patersen Graph The Petersen graph (10 vertices and 15 edges) does not have Hamiltonian circuits. Why? Split the edges into three groups: external (1-2, 2-3, 3-4, 4-5, 5-1), middle (1-6, 2-7, 3-8, 4-9, 5-10), internal (6-8, 8-10, 10-7, 7-9, 9-6). A Hamiltonian circuit, comprising 10 edges, contain at least 4 edges in one of the groups. External group : not possible to have 5 HC edges not possible to have 4 HC edges, as a HC cannot exist (see the figure). 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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4 ENGG1007 FCS Patersen Graph (continue) Patersen Graph (continue) Middle group : not possible to have 5 HC edges not possible to have 4 HC edges, as a HC cannot exist (see the figure). Internal group: not possible to have 5 edges not possible to have 4 HC edges, as a HC cannot exist (see the figure). 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10
5 ENGG1007 FCS Word Puzzle Word Puzzle Let this list of words HEN, HUT, WIT, SAW, CAR, CUB, MOB, DIM, RED, SON. be replaced HUT and SON by HOT and SUN. Then we can arrange the words in a closed chain, i.e, there exists a Hamiltonian circuit for the graph on the right.

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6 ENGG1007 FCS Hamiltonian Paths and Circuits Hamiltonian Paths and Circuits A Hamiltonian path Hamiltonian path is a path containing every vertex once and only once. A Hamiltonian circuit Hamiltonian circuit is a circuit containing every vertex once and only once. Example: (round-the-world puzzle)
7 ENGG1007 FCS Existence of Hamiltonian Paths Existence of Hamiltonian Paths Unlike Euler paths/circuits, there is no simple condition to check whether a graph has a Hamiltonian path/circuit.

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13Hamiltonian_10_v2 - ENGG1007 Foundations of Computer...

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