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.
Tucker, Applied Combinatorics, Section 2.2, Tamsen Hunter
Definition of Hamilton Path: a path that touches every vertex at most once.
j2.2 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
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
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
Equation in Math Type