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

 Page 3 / 4
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. Ao's counterexample for the case k=4

## 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.

#### Questions & Answers

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get Jobilize Job Search Mobile App in your pocket Now!

Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications? By OpenStax By Brooke Delaney By Anindyo Mukhopadhyay By By By JavaChamp Team By Candice Butts By Tony Pizur By Cameron Casey By OpenStax