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2.2 Hamilton Circuits. Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter. Hamilton Circuits Hamilton Paths. 2.2 Hamilton Circuits. a. b. d. c. Definition of Hamilton Path : a path that touches every vertex at most once. a. b. d. c. 2.2 Hamilton Circuits.

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2 2 hamilton circuits

2.2 Hamilton Circuits

Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter

Hamilton Circuits

Hamilton Paths

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2.2 Hamilton Circuits

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Definition of Hamilton Path: a path that touches every vertex at most once.

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2.2 Hamilton Circuits

Definition of Hamilton Circuit: a path that touches every vertex at most once and returns to the starting vertex.

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2.2 Building Hamilton Circuits

Rule 1: If a vertex x has degree 2, both of the edges incident to x must be part of a Hamilton Circuit

The red lines indicate the vertices with degree two.

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2.2 Hamilton Circuit

Rule 2: No proper subcircuit, that is, a circuit not containing all vertices, can be formed when building a Hamilton Circuit

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2.2 Hamilton Circuit

Rule 3: Once the Hamilton Circuit is required to use two edges at a vertex x, all other (unused) edges incident at x can be deleted.

The red lines indicate the edges that have been removed.

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2.2 Hamilton Circuits

Applying the Rules One & Two

Rule One: a and g are vertices of degree 2, both of the edges connected to those 2 vertices must be used.

Rule Two: You must use all of the vertices to make a Hamilton Circuit, leaving out a vertex would not form a circuit.

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2.2 Hamilton Circuits

Step One: We have two choices leaving i- ij or ik if we choose ij then Rule Three applies.

Step Two: Edges jf and ik are not needed in order to have a Hamilton Circuit, so they can be taken out.

Step Three: We now have two choices leaving j, jf or jk. If we choose jk, then Rule Three applies and we can delete jf.

Applying the Rule Three

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2.2 Hamilton Circuits

Theorem 1

A connected graph with n vertices, n >2, has a Hamilton circuit if the degree of each vertex is at least n/2

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2.2 Hamilton Circuits

Theorem 2

Let G be a connected graph with n vertices, and let the vertices be indexed x1, x2,…, xn, so that deg(xi)  deg(xi+1). If for each k n/2, either deg (xk) > k or deg(xn+k)  n – k, then G has a Hamilton circuit

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2.2 Hamilton Circuits

Theorem 3

Suppose a planar graph G has a Hamilton circuit H. Let G be drawn with any planar depiction, and let ri denote the number of regions inside the Hamilton circuit bounded be i edges in this depiction. Let r´i be the number of regions outside the circuit bounded by i edges. Then the numbers ri and r´i satisfy the equation

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2.2 Hamilton Circuit
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2.2 Hamilton Circuits

Equation in Math Type

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2.2 Hamilton Circuit

Theorem 4

Every tournament has a Hamilton path.

A tournament is a directed graph obtained from a complete (undirected) graph by giving a direction to each edge.

All of the tournaments for this graph are;

a-d-c-b, d-c-b-a, c-b-d-a, b-d-a-c, and d-b-a-c.

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One answer is;

a-g-c-b-f-e-i-k-h-d-j-a

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2.2 Hamilton Circuits

Class Work

Exercises to Work On (p. 73 #3)

Find a Hamilton Circuit or prove that one doesn’t exist.

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a-f-b-g-c-h-d-e-a is forced by Rule One, and then forms a subcircuit, violating Rule Two.

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2.2 Hamilton Circuits

Class Work Exercises to Work On

Find a Hamilton circuit in the following graph. If one exists. If one doesn’t then explain why.