I EXAMPLE 10 Let D, = (r. f\ r" = f² = e, rf = fr¯!). Then a Hamiltonian circuit in Cay({(r, 0), (f, 0), (e, 1)}:D, Z) with m even is traced in Figure 30.8. The sequence of generators that traces the circuit is m * [(n – 1) * (r, 0), (f, 0), (n – 1) * (r, 0), (e, 1)]. (F. 0) (e, 0)* (F. 1) (e, 1) (rf. 0) (r, 0) (r, 1) ..iterate (Ff. 0) (r, 0) (*, 1) (p-1f. 0) • (r-! 0) (p-1f, 1) (-1, 1)
. Let Dn be as in Example 10. Find a Hamiltonian circuit in
Cay({(r, 0), (f, 0), (e, 1)}:D4 ⨁ Z5).
Does your circuit generalize to the case Dn ⨁ Zn+1 for all n ≥ 4?
![I EXAMPLE 10 Let
D, = (r. f\ r" = f² = e, rf = fr¯!).
Then a Hamiltonian circuit in
Cay({(r, 0), (f, 0), (e, 1)}:D, Z)
with m even is traced in Figure 30.8. The sequence of generators that
traces the circuit is
m * [(n – 1) * (r, 0), (f, 0), (n – 1) * (r, 0), (e, 1)].
(F. 0)
(e, 0)*
(F. 1)
(e, 1)
(rf. 0)
(r, 0)
(r, 1)
..iterate
(Ff. 0)
(r, 0)
(*, 1)
(p-1f. 0) •
(r-! 0)
(p-1f, 1)
(-1, 1)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fddca77dd-1cf4-407d-bd01-1b004861ab6e%2F43bd79b5-84b3-4079-811c-8a5e661e06f0%2Far26umk.png&w=3840&q=75)
Introduction:
A Hamiltonian circuit is a circuit in a graph that visits every vertex exactly once and ends at the starting vertex. In other words, it is a closed path that includes all the vertices of the graph.
More formally, a Hamiltonian circuit is a simple cycle that contains all the vertices of a graph G. If such a cycle exists in G, then G is said to be Hamiltonian.
The concept of a Hamiltonian circuit is named after the Irish mathematician Sir William Rowan Hamilton, who was interested in finding a path in a dodecahedron that visits each vertex exactly once.
The problem of determining whether a graph has a Hamiltonian circuit is known as the Hamiltonian circuit problem, and it is one of the most famous NP-complete problems in computer science and mathematics. That is, it is a problem that is believed to be computationally intractable, meaning that there is no known efficient algorithm for solving it for all instances.
Finding a Hamiltonian circuit in a graph is a challenging task, and there is no general algorithm that can solve it for all graphs. However, there are several techniques and heuristics that can be used to find Hamiltonian circuits in some special cases or in graphs with certain properties.
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