The document discusses Hamiltonian cycles in graphs. It defines a Hamiltonian cycle as a cycle in a graph that visits each vertex exactly once. It provides the history of Hamiltonian cycles stemming from a game invented by William Rowan Hamilton. It distinguishes Hamiltonian cycles from paths and discusses properties like every vertex must have a degree of at least 2. It also presents a naive algorithm to find Hamiltonian cycles by checking all permutations of vertices and analyzes its exponential time complexity, showing the problem is NP-complete.

1. HAMILTONIAN
CIRCUIT
- William Rowan Hamilton

2. Content :
1. Hamiltonian Cycles Brief History ,definition.
2. Diference between path and cycle.
3. Few Interesting condition.
4. Naive Algorithm and its time complexity .

3. Hamiltonian Cycles History :
● Icosian game invented by Sir. William Rowan Hamilton (1805–1865) in
1857 and sold to a London game dealer in 1859 for 25 pounds.
● Game Description: The corners of a regular dodecahedron are labeled
with the names of cities; the task is to find a circular tour along the edges
of the dodecahedron visiting each city exactly once.
.
● Solution Model: Hamiltonian cycle; i.e. look for a cycle in the
corresponding dodecahedral graph which contains each vertex exactly
once. The Platonic Solid used in Icosiangame; the corresponding
Hamiltonian cycle is designated by darkened edges.

4. Dodecahedron - 3 dimensional Dodecahedron-2
dimensional

5. THIS IS HAMILTONIAN
CYCLE .
A GRAPH CONTAINS A
CYCLE WHICH HAS
ALL VERTICES OF
GRAPH VISITING
EACH VERTEX ONLY
ONCE .

6. Basic Definition :
● Hamiltonian Cycle: If G = (V, E) is a graph or multi-graph with |V|>=3, we
say that G has a Hamiltonian cycle if there is a cycle in G that contains
every vertex in V exactly once .
● Necessary and Sufficient Conditions: Condition for Hamiltonian based
on Linear Diophantine Equation Systems with Cycle Vector, 2009 3rd Intl.
Conf. on Genetic and Evolutionary Computing .
● Hamiltonian Path: A Hamiltonian path is a path (and not a cycle) in G that
contains each vertex. It is possible, however, for a graph to have a
Hamiltonian path without having a Hamiltonian cycle. Hamiltonian path
having a cycle Hamiltonian cycle .

7. Graph contains Hamiltonian path but not cycle.Exampl
e

8. Some Interesting Facts about Hamiltonian cycle :
● Every complete graph with more then two vertices is hamiltonian graph
.
Definition of complete graph: In a undirected simple graph every pair
of a graph is connected with a unique pair.
● If G=(V,E) has a hamiltonian cycle then for all v ∈ V , deg(v)>=2.
● If a ∈ V and deg(a)=2 then two edge incident with vertex a must appear
in every hamiltonian cycle.
● There is no know particular condition which are both necessary and
sufficient.

9. ● So we can define the hamiltonian-cycle problem, “Does a graph G have
a hamiltonian cycle?” as a formal language:
“A graph contains hamiltonian cycle if it is hamiltonian graph .”
● So how might an algorithm decide the language HAM-CYCLE?
Naive - Solution :
one possible decision algorithm lists all permutations of the vertices
of G and then checks each permutation to see if it is a hamiltonian path.
What is the running time of this algorithm? If we use the “reasonable”
encoding of a graph as its adjacency matrix, the number m of vertices in
the graph is Ω√ n , where n= length of the encoding of G. There are m!
possible permutations of vertices ,and therefore the running time is

10. Ωm! =Ω(√ n !)=Ω(2√ n),
which is not O(nk )for any constant k. Thus, this naive algorithm does not run
in polynomial time. In fact, the hamiltonian-cycle problem is NP-complete,

11. Verification
Algorithms
Anshika Pandey
205119018

12. Verification Algorithms
● What is verification of Algorithms?
● Verification algorithm: two argument algorithm A where x is ordinary string and y
is a binary string called certificate.
Hamiltonian Algorithm
Given: Vertices in order. And a graph G such that G is a hamiltonian.
Proof: Check whether it is a permutation of vertices of V, and whether each of the edge
exists in the graph. We can do this in O(n^2) time where n is length of encoding of G.
Thus, a proof that a hamiltonian cycle exists in a graph can be verified in
polynomial time.

13. Complexity Class NP
● What is NP complexity class
Class of language that can be verified in a polynomial time algorithm.
● What is P class?
Problems that are solvable in polynomial time, i.e. these problems can be
solved in time O(n^k) in worst-case, where k is constant.
● Hamiltonian Cycle is a NP complete problem.
● If L belongs to P then L belongs to NP. P is a subset of NP.
● Areas of Research
It is not known whether P=NP.
It is also unknown that whether NP is closed under complement i.e if L belongs to
NP does it imply complement(L) belongs to NP?

14. These are four possibilities for relationship among complexity classes.

15. PSEUDO CODE
Arjun Singh
205119020

16. Algorithm Hamiltonian(k)
{
do
{
NextVertex(R);
if(x[k]==0)
return ;
if(k==n)
print(x[1:n]);
else
Hamiltonian(k+1);
}
while(true);
}

17. Algorithm NextVertex(k)
{
do
{
x[k]=(x[k]+1)mod(n+1);
if(x[k]==0)
return ;
if(G[k[k-1],x[k] !=0)
{
for j=1 to k-1 do
if(x[j] == x[k])
break;
if(j==k)
if(k<n or (k==n)
&& G[x[n],x[1]] !=0
return ;
}
while(ture);
}

18. Traversing
Saurav Kr Gupta
205119092

19. Graph has Hamiltonian Cycles:
1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 1
1 -> 2 -> 6 -> 5 -> 4 -> 3 -> 1
1 -> 6 -> 2 -> 5 -> 4 -> 3 -> 1
…
2 -> 3 -> 4 -> 5 -> 6 -> 1 -> 2

21. Pendant Vertices
Articulation Point

22. 0 1 1 0 1
1 0 1 1 1
1 1 0 1 0
0 1 1 0 1
1 1 0 1 0
Adjacency Matrix
X
1 2 3 4 5