The document summarizes research on Hamiltonian graphs. It begins by describing Sir William Hamilton's 18th century mathematical game that inspired the concept of Hamiltonian graphs. It then provides definitions and theorems related to Hamiltonian graphs, including Dirac's theorem from 1952 and Chvátal's theorem from 1972, which provide necessary conditions for a graph to be Hamiltonian based on the degrees of its vertices. The document concludes that while these theorems are useful for determining whether graphs are Hamiltonian, they are not sufficient tests on their own as some graphs satisfy the theorems but are non-Hamiltonian, or vice versa.

1. International Reseach Journal,November,2010 ISSN-0975-3486 RNI: RAJBIL 2009/300097 VOL-I *ISSUE 14
64 RESEARCH ANALYSIS AND EVALUATION
Research Paper
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November, 2010
1.1INTRODUCTION:
In1859,SirWilliamHamiltondevisedamath-
ematical game on the graph of dodecahedron i.e.
puzzle.Inthispuzzle20verticesofdodecahedronwas
labelled by the name of some city and the objective
of puzzle was construct a tour of all the cities going
throughedgessuchthatnocityappearmorethanone
in the tour. That means one needs to construct a
closed walk that each vertex graph is traced exactly
once. This closed walk is called Hamiltonian circuit
and a graph containing Hamiltonian circuit is called
aHamiltoniangraph.
STUDYOFDIRAC & CHARVALTHEOREM
FOR HAMILTONIAN GRAPH
* Dr. Sudhir Prakash Srivastava
them,eachoflengthatlast2.Thegraphabovefig.2.1
is an example of Theta graph is clear that theta graph
is obviously non-hamiltonian and every non-
homiltonian connected graph has theta subgraph.
Theorem 2.1 :- The complete graph Kn (n³3) is
Hamiltonian.
DuringamathematicalgameongraphofSirWillianHamiltonin1859giveaconceptofHamiltoniangraph.
SeveralnecessaryandsufficientconditionforHamiltoniangraphexist,butnoelegantcharacterizationof
Hamiltonian graph is known. In fact, the problem of determining which graph are Hamiltonian as one the
major problem of graph theory. In this paper we try to determine different graph whether they are
Hamiltonian or not with help of Dirac & Charval Theorem.
A B S T R A C T
1
14
13
12
5
8
9
10
15
20
16
19
18
17
4
3
2
6
7
11
Fig 1.1 : Dodecahedron
In other word a spanning cycle in a graph is
calledHamiltoniancircuitorcycleandagraphhaving
aHamiltoniancyclecalledHamiltoniangraph.Several
necessary and sufficient conditions for Hamiltonian
graph exist but no elegent characterization of Hamil-
tonian graph is known.
2.1Somebasicdefinitionandtheorem
Definition2.1: Ablockwithtwononadjacentvertex
of degree 3 and all other vertices of degree 2 is called
Theta graph.
Fig2.1
ThisimpliesthatThetagraphconsistoftwo
vertices of degree 3 and three disjoining path join
4
6
3
1 2
5
7
Theorem 2.2 :- The complete bipartite
graph Km, n
is Hamiltonian iff m=n and
n>1.
Theorem 2.3:-EveryHamiltoniangraph
is 2-connected. Let G be a Hamiltonian
graphandZbeaHamiltoniancircuitinG.
For any vertex v of G, Z-v is connected
and hence G-v also connected. Hence G
has no cut set and thus G is 2-connected.
Theorem 2.4:-EveryGraphGhasaHamiltonianifthe
sumofdegreeofeverypairverticesVi,VjinGsatisfies
the condition d(Vi
)+ d(Vj
) ³ n-1.
Theorem 2.5 :- Let the number of edges of G be m.
Then G has Hamiltonian circuit if m
³
21
( 3 6)
2
n n− − wherenisthenumberofvertices.
Let u and v are any two vertex of G that are not
adjacent. Suppose H be a graph produced by elimi-
natinguandvfromGalongwithanyedgesthathave
u or v as end point. Then H has n-2 vertices and m-
* IET, Dr. R.M.L.Avadh University, Faizabad-224 001

2. International Reseach Journal,November,2010 ISSN-0975-3486 RNI: RAJBIL 2009/300097 VOL-I *ISSUE 14
65RESEARCH ANALYSIS AND EVALUATION
deg(u)-deg(v)edges.Themaximumnumberofedges
that H is n-2
C2
.This happens when there is an edges
connecting every distinct pair of vertices. Thus the
numher of edges of H is at must.
or
21
( 5 6 )
2
n n− +
We have
d e g ( ) d e g ( )m u v− −
³
21
( 5 6)
2
n n− +
Therefore
deg( ) deg( )u v+ ³
21
( 5 6)
2
m n n− − +
by the hypothesis of the theorem
deg( ) deg( )u v+ ³
2 21 1
( 3 6) ( 5 6)
2 2
n n n n− + − − +
= n
Hence proved
The following theorem gives a simple and
useful necessary condition for Hamiltonian graph.
Theorem 2.6 :- If G is Hamiltonian, then for every
nonemptypropersubsetSofV(G),W(G-S)£|S|where
W(H) denotes the numbers of components in any
graph H. Let Z be a Hamiltonian circuit of G.& S be
another proper subset of V(G).
Now W(Z-S) £ |S|. Also Z-S is a spanning
subsetofG-S&W(G-S)£W(Z-S)HenceW(G-S)£|S|.
Note2.1Theabovetheoremisusefulinshowingthat
some graph are non Hamiltonian. For example, con-
siderthecompletebipartitegraphKm,n
withm<n.Let
( 1v , 2v ) be a bipartite of the graph with | 1v |=m and
| 2v |=n. The graph Km,n
- 1v is totally disconnected
graph with n point
Hence w K Vm n , − 1d i =n>m=|
1v
|.
So Km,n
is nonhamiltonian.
NOTE2.2:-Theconverseoftheabovetheoremisnot
condition of the theorem but is non Hamiltonian.
Fig2.2:PeterseGraph
3. Dirac Theorem, 1952
Statement : If G is a graph with n³3 vertices and d³n/
2thenGisHamiltonian.
Proof : Let the above statement is false. Let G be
maximal(Duetoedges)nonHamiltoniangraphwith
n vertex & d³n/2
Since n³3, G cannot be complete
Suppose u and v be nonadjacent vertices in G. By the
choices of G, G+uv is Hamiltonian. Since G is non
HamiltoniancircuitofG+uvmustcontainthelineuv
Thus G has a spanning path v1
, v2
, .............vp
with
origin u=v1
and terminal v=up
.
Suppose S={vi
/ uvi+1 ∈E} and T= { vi
/ i < n and vi
v
∈
E } where E is edge set of G. Clearly up
Ï S ÈTand
hence | S È T| < n ............(1)
Again if vi
∈
S Ç T then v1
v2
.....vi
vp
vp-1
.....vi+1
v1
Is
2
3
5
4
b
c
d
e
1
a
true For example the Petersen graph satisfies the
aHamiltoniancircuitinG.
HenceSÇT=ØSothat|SÇT|=0 ………….(2)
AlsobythedefinitionofSandTd(u)=|S| andd(v)=|T|
Hence by (1) and (2) d(u)+d(v)=|S|+|T|
=|S È T|< n
Thus d(u)+d(v) < n
But since d ³ n/2 we have d(u)+d(v) ³ n which give a
contradiction.
Hence theorem prove.
Lemma:LetGbeagraphwithnpointandu;vbenon
adjacent point in G such that d(u)+d(v) ³ n Then G is
HamiltonianifG+uvisHamiltonian.IfGisHamilto-
nian, then G+uv is also Hamiltonian conversely let
G+uv is Hamiltonian, but G is not.
Then as the proof above theorem we obtain
d(u)+d(v)<n, contradicts the hypothesis d(u)+d(v) ³
n Thus G+uv is Hamiltonian ie. G is Hamiltonian.
According to above lemma we find follow-
ing definition of closure.
Definition 3.1 :- The closure of graph G with n point
is the graph obtained from G by repeatedly joing pair
of nonadjacent vertices whose degree sum is at least
nunitnosuchpairremain.TheclosureofGisdenoted
byc(G)
If two persons construct a closure graph then they
ending sane graph.
Theorem3.1:-c(G)iswelldefined.C(G)iscomplete

3. International Reseach Journal,November,2010 ISSN-0975-3486 RNI: RAJBIL 2009/300097 VOL-I *ISSUE 14
66 RESEARCH ANALYSIS AND EVALUATION
withfollowingexample
c(G).
Clearly|S|=çSç=n-1-
'
( )d v &çTç=n-1-
'
( )d u
......................(3).
Also by the choice of u and v, each vertex
in S has degree at most '
( )d u and each vertex in
TU{u}hasdegreeatmost '
( )d v .Putting(2)inthe
firstequationof(3)weget+|S|>p-1-(p-m)=m-1.
Hence |S| ³ m. Hence c(G) has at least m point with
degree £m. ...................(4)
From(3),|T|=n-1-m.SinceeachvertexinT
U {u} has degree, this implies that c (G) has at least
n-m vertices of degree <p-m ...................(5)
BecauseGisaspanningsubgraphofc(G),degreeof
each point in G cannot exceed that in c(G). hence
statement (4) and (5) hold in the case of G also.
Hence md <mand n md − <n-m.alsoby(1)&(2)
m < n/2. This is contradicts the hypothesis on G.
c(G) is complete. Hence G is Hamiltonian.
5.Conclusion:-
According to unit 3 and 4 it is clear that
Dirac'sandchvatal'stheoremplayonimportantroll
to decide any graph is Hamilton or not. But both are
theorem is only hypothesis because some graph,
which is not a hamiltonian graph satisfy the law of
above theorem. Some graph not satisfy the law of
above theorem but they are hamiltonian. In general
way by fidning the degree sequence & appying the
above both theorem, after we can deduce that given
graph is hamiltonian. Any graph that satisfies the
hypothesis of Dirac theorem also satisfies the hy-
pothesisofchvatal'stheorem.Hencechvatal'stheo-
rem is stronger than Dirac's theorem.
However the graph G in fig 3.1 has degree
of sequence 2, 2, 3, 3, 3, 3 and it does not satisfy the
hypothesis of Chvatal's theorem.
Fig3.1
Theorem 3.2 :-AgraphisHamiltonianiffitsclosure
isHamiltonian.
Lemma3.2:-LetGbeagraphwithatlest3pointifc(G)
is complete, then G is Hamiltonian. The above theo-
rem and Lemma are useful in showing that a given
graphisHamiltonian.
Forexamplefraphinfig3.1isHamiltonian.
G isHamiltonian.Henced>n/2and n>3=GisHamil-
tonian.
4.ChauatalTheoraem1972
In 1972 a mathematician chauatal proved a theorem
for test he given graph is Hamiltonian or not.
Statement :- Let G be a graph with degree sequence
(d1
d2
.....dn
) where d1
£ d2
£ d3
£ ........ £ dn
and n ³ 3.
Suppose that for every value of m less than n/2 either
dm
> m or dn–m
> n–m (i.e. there is no value of m less
than n/2 for which dm
£ m and dn–m
< n–m. Then G is
Hamiltonian.
Proof :- yet G satisfy the hypothsis of the theorem .
We consider c(G) is completed, suppose we denote
the degree of a vertex v in c(G) by d1
(v).
If the possible, let c(G) be not complete .
Now let u and v be two nonadjacent vertices in c(G)
with
' '
( ) ( )d u d v≤ .....................(1)
and '
( )d u + '
( )d v as large as possible. Let
'
( )d u =m.
Since no two nonadjacent point in c(G) can have
degree sum n or more, we have '
( )d u +
'
( )d v <
n
'
( )d v < n- '
( )d u '
( )d v <n-m.........................(2)
Let S-denote the set of vertices in V-
{v}whicharenoadjacenttovinc(G).LetTdenotethe
set of vertices inV-{u} which are not adjacent to u in
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