GATE CSE
First time here? Checkout the FAQ!
x
+3 votes
102 views

Let G be a graph with 10 vertices and 31 edges. If G has 3 vertices of degree 10, 1 vertex of degree 8 and 2 vertices of degree 5 and the other four vertices of degree at least 3, how many vertices are of degree 3________?

my solution:

Σ deg(v) = 2|E|

3*10 + 1*8 + 2*5 + 4*(>=3) = 2*31

4*(>=3) = 62- (30+8+10) = 14

I think we can have 3 vertices each of degree 3 and vertex of degree 5, so my answer is 3 but given answer is 2.

Given answer: 2

asked in Graph Theory by Veteran (15.4k points)   | 102 views
3 vertex of degree 10 means it is a multigraph...in simple graph degree cannot be greater than 9...or this que may be framed wrongly
multigraph also follows this property:
Σ deg(v) = 2|E|

 

what's wrong?

1 Answer

+5 votes
Best answer
ohk,

sum of degree  = 2 * edges

3 * 10 +  1 * 8 + 2 * 5  + x1 + x2 + x3  + x4 =  2 * 31

x1 + x2 + x3 + x4 =   62 - 48

x1 + x2 + x3 + x4 = 14

each node have degree atleast three means  

x1 + x2 + x3 + x4 =   14 - 4 * 3 = 2

x1 + x2 + x3 + x4 = 2         

Means either one node take 2 value , means  one node have degree 3 + 2 :  5 ( Not possible , because their are only 2 nodes of degree 5 )

Or  

two nodes have 1 , 1 value       means two nodes have 4 degree , and left two nodes have 3 degree.

Hence nodes having degree  = 3  are 2.
answered by Loyal (2.8k points)  
selected by
( Not possible , because their are only 2 nodes of degree 5 )  i was thinking the same later because it is mentioned in the question that there 2 vertices of 5 degree.
thanks!

You can also do it like this:

Let 'x' be the vertices of degree 3

So, (4-x) vertices are of degree >=4 (because 4 vertices have degree >=3)

 

So,

3(10) + 1(8) + 2(5) + x(3)  + (4-x)(4) >= 62

$\therefore$  2 >= x

Yes, thats why  I have taken >= 62


Top Users Sep 2017
  1. Habibkhan

    6970 Points

  2. Warrior

    2490 Points

  3. Arjun

    2368 Points

  4. rishu_darkshadow

    2136 Points

  5. A_i_$_h

    2004 Points

  6. nikunj

    1980 Points

  7. makhdoom ghaya

    1760 Points

  8. manu00x

    1750 Points

  9. Bikram

    1744 Points

  10. SiddharthMahapatra

    1718 Points


26,060 questions
33,668 answers
79,747 comments
31,079 users