# 5.1 Vizing's conjecture and related graph properties  (Page 4/4)

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## Proposition 2:

$|G|\le \delta \left(G\right)\Delta \left(G\right)$ .

Proof: We start by looking at the vertex with the smallest degree, ${v}_{s}$ . Since for any vertex $v$ , $N\left(v\right)\cap N\left({v}_{s}\right)\ne \varnothing$ we can say that every vertex is connected to either ${v}_{s}$ or a vertex in $N\left({v}_{s}\right)$ . There are $\delta \left(G\right)$ such vertices, having a maximum degree of $\Delta \left(G\right)$ , implying that there are at most $\delta \left(G\right)\Delta \left(G\right)$ vertices in the graph.

## Proposition 3 (the lonely neighbor property):

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

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

## Proposition 4:

For any edge $\left(u,v\right)\in G$ , 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 $deg\left(u\right)=deg\left(v\right)=3$ . $u$ 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 $\left(u,w\right)\in G$ or $\left(v,w\right)\in G$ . 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 $G\left(V,E\right)$ is a graph in $S$ then there are atleast $\frac{|E|}{6}$ 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 $\frac{1}{6}$ of a ${K}_{4}$ .

## Proposition 6:

For any $v\in V\left(G\right)$ , $v$ is contained in at least $⌈\frac{\text{deg(}v\text{)}}{3}⌉$ ${K}_{4}$ 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 $\frac{\text{deg(}v\text{)}}{3}$ 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 $⌈\frac{\text{deg(}v\text{)}}{3}⌉$ ${K}_{4}$ subgraphs.

## Proposition 7:

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

Proof: Let $\left\{{v}_{1},{v}_{2},{v}_{3},{v}_{4}\right\}\in G$ induce a ${K}_{4}$ subgraph. Suppose for the sake of contradiction that $deg\left({v}_{i}\right)\le 4$ , for $i\in \left\{1,2,3,4\right\}$ . ${K}_{4}$ 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 ( $\left\{{v}_{2},{v}_{3}\right\}$ ) 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 $\left\{{v}_{1},{v}_{2},{v}_{3}\right\}$ would have degree more than 5, so it must have degree 3. However, the graph induced by $\left\{{v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5}\right\}$ 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 $\left\{{v}_{1},{v}_{2},{v}_{3},{v}_{4}\right\}$ , 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.

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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
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Damian
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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