2.2 Hamilton Circuits
Hamilton CircuitsHamilton Paths
Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter
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Definition of Hamilton Path: a path that touches every vertex at most once.
2.2 Hamilton Circuits
2.2 Hamilton Circuits
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Definition of Hamilton Circuit: a path that touches every vertex at most once and returns to the starting vertex.
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 a
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The red lines indicate the vertices with degree two.
2.2 Hamilton CircuitRule 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 CircuitRule 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.
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The red lines indicate the edges that have been removed.
2.2 Hamilton Circuits
Applying the Rules One & Two
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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.
2.2 Hamilton Circuits
Applying the Rule Three
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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.
2.2 Hamilton CircuitsTheorem 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 CircuitsTheorem 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
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
2.2 Hamilton Circuitc
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2.2 Hamilton Circuits
Equation in Math Type
2.2 Hamilton CircuitTheorem 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.
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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.
2.2 Hamilton CircuitsClass Work
Exercises to Work On (p. 73 #3)
Find a Hamilton Circuit or prove that one doesn’t exist.
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One answer is;
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2.2 Hamilton CircuitsClass Work Exercises to Work On
Find a Hamilton circuit in the following graph. If one exists. If one doesn’t then explain why.
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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.