International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
Finding the Number of Hamiltonian Cycles in a Complete
Graph
T. Ravi Kumar
Associate Professor
in Computer Science
K. L. University,
Vaddeswaram, Guntur District,
India
S. Venu Madhava Sarma
Assistant Professor
of Mathematics,
K. L. University,
Vaddeswaram, Guntur District,
India
T. V. Pradeep Kumar,
Assistant Professor
of Mathematics,
A.N.U College of Engineering &
Technology,
Acharya Nagarjuna University.
ABSTRACT
This paper seeks to review some ideas and results relating to Hamiltonian graphs. Every edge in a cubic graph lies on an
even number of Hamiltonian Cycles, and every Hamiltonian Cubic Graph contains at least three Hamiltonian cycles.
Finally G be a Complete Graph, all of whose vertices have odd degree. Then every edge is contained in an even number
of Hamiltonian cycles.
Key words : Graph, Euler Graph, Hamiltonian graph, Complete Graph, Regular Graph
1. Introduction:
The origin of graph theory started with the problem of Koinsber bridge, in 1735.
This problem lead to the concept of Eulerian Graph. Euler studied the problem of Koinsberg bridge and
constructed a structure to solve the problem called Eulerian graph.
In 1840, A.F Mobius gave the idea of complete graph and bipartite graph and Kuratowski proved that they are
planar by means of recreational problems.
The concept of tree, (a connected graph without cycles was implemented by Gustav Kirchhoff in 1845, and
he employed graph theoretical ideas in the calculation of currents in electrical networks or circuits.
In 1852, Thomas Gutherie found the famous four color problem.
Then in 1856, Thomas. P. Kirkman and William R.Hamilton studied cycles on polyhydra and invented the
concept called Hamiltonian graph by studying trips that visited certain sites exactly once.
In 1913, H.Dudeney mentioned a puzzle problem. Eventhough the four color problem was invented it was
solved only after a century by Kenneth Appel and Wolfgang Haken.
This time is considered as the birth of Graph Theory.
Caley studied particular analytical forms from differential calculus to study the trees. This had many
implications in theoretical chemistry. This lead to the invention of enumerative graph theory.
Any how the term “Graph” was introduced by Sylvester in 1878 where he drew an analogy between “Quantic
invariants” and covariants of algebra and molecular diagrams.
In 1941, Ramsey worked on colorations which lead to the identification of another branch of graph theory
called extremel graph theory.
In 1969, the four color problem was solved using computers by Heinrich. The study of asymptotic graph
connectivity gave rise to random graph theory.
In 1971 R.Halin introduced an example of minimally 3- connected Graphs.
1.1 Definition: A graph – usually denoted G(V,E) or G = (V,E) – consists of set of vertices V together with a
set of edges E. The number of vertices in a graph is usually denoted n while the number of edges is usually
denoted m.
1.2 Definition: Vertices are also known as nodes, points and (in social networks) as actors, agents or players.
1.3 Definition: Edges are also known as lines and (in social networks) as ties or links. An edge e = (u,v) is
defined by the unordered pair of vertices that serve as its end points.
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T. Ravi Kumar, S. Venu Madhava Sarma, T. V. Pradeep Kumar
International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
1.4 Example: The graph depicted in Figure 1 has vertex set V={a,b,c,d,e.f} and edge set
E = {(a,b),(b,c),(c,d),(c,e),(d,e),(e,f)}.
d
e
a
f
b
c
Figure 1.
1. 5 Definition: Two vertices u and v are adjacent if there exists an edge (u,v) that connects them.
1.6 Definition: An edge (u,v) is said to be incident upon nodes u and v.
1.7 Definition: An edge e = (u,u) that links a vertex to itself is known as a self-loop or reflexive tie.
1.8 Definition: Every graph has associated with it an adjacency matrix, which is a binary nn matrix A in
which aij = 1 and aji = 1 if vertex vi is adjacent to vertex vj, and aij = 0 and aji = 0 otherwise. The natural
graphical representation of an adjacency matrix is a table, such as shown below.
a b c d e f
a 0 1 0 0 0 0
b 1 0 1 0 0 0
c 0 1 0 1 1 0
d 0 0 1 0 1 0
e 0 0 1 1 0 1
f 0 0 0 0 1 0
Adjacency matrix for graph in Figure 1.
1.9 Definition: Examining either Figure 1 or given adjacency Matrix, we can see that not every vertex is
adjacent to every other. A graph in which all vertices are adjacent to all others is said to be complete.
1.10 Definition: While not every vertex in the graph in Figure 1 is adjacent, one can construct a sequence of
adjacent vertices from any vertex to any other. Graphs with this property are called connected.
1.11 Note: Reachability. Similarly, any pair of vertices in which one vertex can reach the other via a sequence
of adjacent vertices is called reachable. If we determine reachability for every pair of vertices, we can
construct a reachability matrix R such as depicted in Figure 2. The matrix R can be thought of as the result of
applying transitive closure to the adjacency matrix A.
d
e
a
f
b
c
g
Figure: 2
1.12 Definition : A walk is closed if vo = vn.degree of the vertex and is denoted d(v).
1.13 Definition : A tree is a connected graph that contains no cycles. In a tree, every pair of points is
connected by a unique path. That is, there is only one way to get from A to B.
1.14 Definition: A spanning tree for a graph G is a sub-graph of G which is a tree that includes every vertex
of G.
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T. Ravi Kumar, S. Venu Madhava Sarma, T. V. Pradeep Kumar
International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
1.15 Definition: The length of a walk (and therefore a path or trail) is defined as the number of edges it
contains. For example, in Figure 3, the path a,b,c,d,e has length 4.
Figure 3: A labeled tree with 6 vertices and 5 edges
1.16 Definition: The number of vertices adjacent to a given vertex is called the degree of the vertex and is
denoted d(v).
1.17 Definition : In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose
vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V;
that is, U and V are independent sets. Equivalently, a bipartite graph is a graph that does not contain any oddlength cycles.
Figure 4: Example of a bipartite graph.
1.18 Definition : Regular Graph : A graph is regular if all the vertices of G have the same degree. In
particular, if the degree of each vertex is r, the G is regular of degree r.
1.19 Example:
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T. Ravi Kumar, S. Venu Madhava Sarma, T. V. Pradeep Kumar
International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
1.20 Definition : An Eulerian circuit in a graph G is circuit which includes every vertex and every edge of G.
It may pass through a vertex more than once, but because it is a circuit it traverse each edge exactly once. A
graph which has an Eulerian circuit is called an Eulerian graph. An Eulerian path in a graph G is a walk which
passes through every vertex of G and which traverses each edge of G exactly once
1.21 Example : Königsberg bridge problem: The city of Königsberg (now Kaliningrad) had seven bridges on
the Pregel River. People were wondering whether it would be possible to take a walk through the city passing
exactly once on each bridge. Euler built the representative graph, observed that it had vertices of odd degree,
and proved that this made such a walk impossible. Does there exist a walk crossing each of the seven bridges
of Königsberg exactly once?
Figure 5: K¨onigsberg problem
Theorem 1: Every edge in a cubic graph lies on an even number of Hamiltonian Cycles.
Proof: We know doodling on a diagram similar to Figure 6, it is easy to prove the Corollary below.
The "worst" case of the two cycles through e still allows another cycle through f.
Figure 6
Corollary : Every Hamiltonian cubic graph contains at least three Hamiltonian cycles.
In 1975, Sheehan [30] conjectured that every Hamiltonian 4-regular graph has at least one more Hamiltonian
cycle.
63
T. Ravi Kumar, S. Venu Madhava Sarma, T. V. Pradeep Kumar
International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
Theorem 2.Let G be a Complete Graph, all of whose vertices have odd degree. Then every edge is contained
in an even number of Hamiltonian cycles.
Proof :. We illustrate the method first on cubic graphs.
Suppose that G is a Complete Hamiltonian cubic graph.
Let H = (x1, x2, ... , xn) be a Hamiltonian cycle in G.
We now form a new graph „g „whose vertices are Hamiltonian paths starting with the edge x 1,x2.
Before describing adjacencies in‟g‟ , consider the Hamiltonian path
H1 = x1 ,x2 , .... , Xn
Figure 7
Now in G we know that xn is adjacent to Xl
Since G is cubic, Xnis adjacent to another vertex, xr say.
Consider H2, the Hamiltonian path xl ,x2, ….. , xr xn,xn-1,….... , Xr+I
We show this in Figure 7(b).
Now in G we know that xr+1 is adjacent to xr and Xr+2
Since G is cubic, xr+1 is also adjacent to another vertex xS
Consider H3, the Hamiltonian path xI, x2, ... , xS, xr+Ixr+2... , xs
where we do not define “ s' “ precisely. We show H3 in Figure 7(c).
The process described above will continue until we have no new edges of the form xnxr, xr+lxs to add to the
Hamiltonian paths.
To define adjacency in „g‟ we say that HiHj is an edge of „g‟ if we can go from one path Hi to the other Hj by
removing one edge (such as xr xr+1' xsxt) and adding another (such as xnxr, xr+lxs).
The important thing to notice now is that deg HI = 1 and deg Hi = 2, unless Hi is at
the end of the chain started in Figure 7, when deg Hi = 1.
Since „g‟is a graph it must have an even number of vertices of degree 1.
Suppose two of these are HI and Hm .
What stops Hm from having a neighbour in „g‟other than Hm.
It must be that there is an edge from the end vertex (not XI) to Xl
Hence Hm plus this edge is a Hamiltonian cycle.
So, every component of ‘g’ is either a cycle, corresponding to no Hamiltonian cycles in G, or a path,
corresponding to two Hamiltonian cycles through x I, x2 in G.
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T. Ravi Kumar, S. Venu Madhava Sarma, T. V. Pradeep Kumar
International Journal of Engineering Technology, Management and Applied Sciences
www.ijetmas.com November 2015, Volume 3, Special Issue, ISSN 2349-4476
Smith's result therefore follows for cubic graphs.
If G is now regular of degree r where r is odd, then deg H, = r - 2.
This is because xn is adjacent to x1xn-1, and r - 2 other vertices.
It then follows that the vertices of „g‟ all have degree r - 2 or r - 1.
But there have to be an even number of vertices of degree r- 2.
All of these correspond to Hamiltonian cycles through x1,x2 and the general result follows.
Note that the odd parity of r is essential for Thomassen's argument to succeed.
There is some feeling though that the result should be true for even r too.
Thomassen [4] has been able to prove that a second Hamiltonian graph exists if the degree of regularity is
sufficiently large.
This is, of course, not as strong as Thomason's result which tells us that two Hamiltonian cycles exist through
every edge.
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Venu Madhava Sarma.S and T.V. Pradeep Kumar International Journal of Mathematical Archive-2(12),
2011,Page 2538-2542.
Venu Madhava Sarma.S International Journal of Computer Application, Issue 2, Volume 1(February 2012) , Page
21-31.
Venu Madhava Sarma.S and T.V.Pradeep Kumar International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN
0973-9424, Vol. 6 No. III (May, 2012), pp. 47-54
Venu Madhava Sarma.S and T.V.Pradeep Kumar proceedings of Two Day UGC National seminar on “Modern
Trends in Mathematics and Physical Sciences ( NSMTMPS – 2012) dated 20th , 21st Jan, 2012.
Necessary and Sufficient Condition on Line Graphs. Proceedings International Conference on Innovative
Research in Engineering Science and Management published at Tata Mc Grah Hills Company
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