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Proposition 2:

| G | δ ( G ) Δ ( G ) .

Proof: We start by looking at the vertex with the smallest degree, v s . Since for any vertex v , N ( v ) N ( v s ) we can say that every vertex is connected to either v s or a vertex in N ( v s ) . There are δ ( G ) such vertices, having a maximum degree of Δ ( G ) , implying that there are at most δ G Δ ( G ) vertices in the graph.

Proposition 3 (the lonely neighbor property):

For each edge ( u , v ) there exists a vertex w such that exactly one of ( w , v ) or ( w , u ) is in E ( G ) . Assuming ( w , v ) is in E ( G ) then for each vertex y exactly one of ( y , w ) or ( y , u ) is in E ( G ) .

Proof: Let an edge ( u , v ) be given. Because our graph is minimal, there exists a vertex w such that N ( a ) N ( w ) = b for some labels a , b { u , v } . Without loss of generality assign a to u and b to v . Now suppose for the sake of contradiction that ( w , u ) is in E ( G ) . Then { u , v , w } induces a K 3 and so there exists x V ( G ) { u , v , w } such that { u , v , w , x } induces a K 4 , but then x N ( u ) N ( w ) = v a contradiction. Now let a vertex y V ( G ) { u , v , w } be given. Suppose both ( y , w ) and ( y , u ) are in E ( G ) then y N ( u ) N ( w ) = v which is again a contradiction.

Proposition 4:

For any edge ( u , v ) G , at least one of u , v has degree of at least 4.

Proof: Suppose that neither of u nor v have degree of at least 4. It follows from Proposition 1 that d e g ( u ) = d e g ( v ) = 3 . u and v must both be contained in a K 4 subgraph. By the lonely neighbor property, there must exist a vertex w such that either ( u , w ) G or ( v , w ) G . This is a contradiction, as desired.

Proposition 5:

For any induced K 4 subgraph, at most one vertex has degree 3.

Proof: This follows directly from Proposition 4.

Proposition 5:

If G ( V , E ) is a graph in S then there are atleast | E | 6 choices of 4-tuples which induce a K 4 .

Proof: Each edge is in a K 4 and so it must make up at least 1 6 of a K 4 .

Proposition 6:

For any v V ( G ) , v is contained in at least deg( v ) 3 K 4 subgraphs.

Proof: Since any K 4 subgraph that contains v must contain three other vertices, and v must be adjacent to all of these vertices, there must be deg( v ) 3 induced K 4 subgraphs. Since this implies that there may be some uncounted incident edges to v , and that all edges are contained in a K 4 subgraph, it follows that v is contained in at least deg( v ) 3 K 4 subgraphs.

Proposition 7:

In any induced K 4 subgraph, at least one vertex must have a minimum degree of 5.

Proof: Let { v 1 , v 2 , v 3 , v 4 } G induce a K 4 subgraph. Suppose for the sake of contradiction that d e g ( v i ) 4 , for i { 1 , 2 , 3 , 4 } . K 4 is not a minimal 3-cover, so without loss of generality, let v 1 have degree 4. Since all edges are contained in a K 4 subgraph, it follows that two more vertices in our original K 4 subgraph ( { v 2 , v 3 } ) must be part of another K 4 subgraph, along with v 1 , and a fifth vertex, called v 5 . v 4 cannot have degree of 4, since this would imply that at least one of { v 1 , v 2 , v 3 } would have degree more than 5, so it must have degree 3. However, the graph induced by { v 1 , v 2 , v 3 , v 4 , v 5 } is not a minimal 3-cover, so at least one incident edge must be added to v 5 . Call the corresponding adjacent vertex v 6 . Since G is a 3-cover, there must be a vertex that is adjacent to both v 3 and v 6 , which implies that we must add an incident edge to at least one of the vertices in { v 1 , v 2 , v 3 , v 4 } , which leads to a contradiction, as desired.

Conclusion

Future work on this topic could either find a counterexample to the conjecture, or show that there is no counterexample by using bounds to arrive at a contradiction.

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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Ramkumar Reply
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Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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