# 5.1 Vizing's conjecture and related graph properties  (Page 3/4)

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1. In the graph G choose two non-adjacent vertices $u$ and $v$ which have not yet been examined. If all pairs $\left(u,v\right)$ have been looked at then stop.
2. Create the graph ${G}^{\text{'}}$ by adding the edge $\left(u,v\right)$
3. If $\gamma \left({G}^{\text{'}}\right)=\gamma \left(G\right)$ then $G←{G}^{\text{'}}$ , otherwise return to 1 and pick another pair.

It is obvious that after completing the algorithm the graph ${G}^{\text{'}}$ will have the same domination number as $G$ . Next it should be apparent that for any graph $H$ if $D$ dominates $G\square H$ then $D$ dominates ${G}^{\text{'}}\square H$ since ${G}^{\text{'}}\square H$ is just $G\square H$ with extra edges, i.e. $\gamma \left({G}^{\text{'}}\square H\right)\le \gamma \left(G\square H\right)$ . So if Vizing's Conjecture holds for the special case of k-critical then it holds for all graphs, and contrastingly, if there is a counterexample the smallest will be k-critical.

## Our contribution

Sun's proof is cumbersome and broken into many cases, but there is potential to eliminate the case of $p$ and $q$ being adjacent if a certain conjecture from Sumner and Blitch [link] is true. Specifically Sumner and Blitch conjectured that in every k-critical graph $i\left(G\right)=\gamma \left(G\right)$ where $i\left(G\right)$ is the Independence Number. The conjecture has been disproven for $k\ge 4$ but remains open in the case $k=3$ [link] . We've put work into developing a counterexample or showing that none exist, in hopes of shortening Sun's proof. This is equivalent to saying in if $G$ is 3-critical, then the complement of $G$ must have a ${K}_{3}$ subgraph which is not contained in a ${K}_{4}$ subgraph.

## 3-covers

We call a graph $G$ a k-cover if for every k-1-tuple of vertices $\left({v}_{1},...,{v}_{k-1}\right)$ there exists a vertex $w$ such that $w\in \bigcap _{i=1}^{k-1}N\left({v}_{i}\right)$ . K-covers are complements to graphs with domination number k, and specifically we're interested in 3-covers. So to add in the idea of the complement of a k-critical graph, we have come up with the idea of a minimal 3-cover. We call a 3-cover minimal if for every edge $\left(u,v\right)\in E$ there exists $w\in V$ and a choice of labels $a,b\in \left(u,v\right)$ s.t. $N\left(a\right)\cap N\left(w\right)=b$ . The relation between a domination 3 graphs and the minimality of their complement is that a graph $G$ is a 3-edge-critical graph if and only if $\overline{G}$ is a minimal 3-cover.

## Reformulation of the conjecture

We can reformulate the original conjecture in a more constructuve form using 3-covers. An equivalent form is that there is no minimal 3-cover in which every ${K}_{3}$ is contained in a ${K}_{4}$ .

## Work towards a proof of the conjecture

Starting by describing the set of counterexamples to the conjecture, we will want to show either that the set is empty, or produce an element from the set to disprove the conjecture. Consider graphs $G\left(V,E\right)$ from the set

$S=\left\{G:G\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{minimal}\phantom{\rule{4.pt}{0ex}}3\text{-cover}\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\text{every}\phantom{\rule{4.pt}{0ex}}{K}_{3}\subset {K}_{4}\right\}$

## Proposition 1:

$\delta \left(G\right)\ge 3$ .

Proof: Since $G$ has the property of being a minimal 3-cover if follows that $G$ is connected, and so each vertex $v$ has $deg\left(v\right)\ge 1$ . Since $G$ is a 3-cover, each edge is contained in a triangle, otherwise there would be a pair which has no common neighbors (the endpoints of any edge not in a triangle). This means any vertex containing an edge also belongs to a triple inducing a ${K}_{3}$ . By the last property of $G$ we have that each of these triples can be extended to a 4-tuple which induces a ${K}_{4}$ . Since each vertex is inside of such a 4-tuple, it follows that each vertex has degree at least 3.

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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
what is Nano technology ?
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
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
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