2.2 Hamilton Circuits Hamilton Circuits Hamilton Paths Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter

ab cd Definition of Hamilton Path: a path that touches every vertex at most once. 2.2 Hamilton Circuits

ab cd 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 f b h c k i e d j g The red lines indicate the vertices with degree two.

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 c ab d ef g h i

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. a f b h c k i e d j g The red lines indicate the edges that have been removed.

2.2 Hamilton Circuits Applying the Rules One & Two a f b h c k i e d j g 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 a f b h c k i e d j g 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 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 a b c d e

2.2 Hamilton Circuits Theorem 2 Let G be a connected graph with n vertices, and let the vertices be indexed x 1, x 2,…, x n, so that deg(x i )  deg(x i+1 ). If for each k  n/2, either deg (x k ) > k or deg(x n+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 r i 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 r i and r´ i satisfy the equation

2.2 Hamilton Circuit c q a b d e f g h i j k l m no p

2.2 Hamilton Circuits Equation in Math Type

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. a b c d 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 Circuits Class Work Exercises to Work On (p. 73 #3) Find a Hamilton Circuit or prove that one doesn’t exist. b cd a e f g h i j k One answer is; a-g-c-b-f-e-i-k-h-d-j-a

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