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Consider a Hamiltonian Graph $G$ with no loops or parallel edges and with $\left | V\left ( G \right ) \right |= n\geq 3$. The which of the following is true?

  1. $\text{deg}\left ( v \right )\geq \frac{n}{2}$ for each vertex $v\\$
  2. $\left | E\left ( G \right ) \right |\geq \frac{1}{2}\left ( n-1 \right )\left ( n-2 \right )+2 \\$
  3. $\text{deg}\left ( v \right )+\text{deg}\left ( w \right )\geq n$ whenever $v$ and $w$ are not connected by an edge
  4. All of the above
in Graph Theory recategorized
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4 Comments

@deka, debosmita, see my answer
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Debasmita Bhoumik  My initial comment logic was wrong. I edited my comment. 

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take n=4   a square graph . does condition 2 holds here

E(G)=4  , n=4  so 3x2 /2 +2 =5

so E(G) is not greater than or equal to 5

so 2 is not true
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4 Answers

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Yeas all the options are true, in fact some of this are well known theorems

Dirac's Theorem:

if G is a simple graph of n vertices ,where n>=3 ,such that the every vertex of G has a degree at least n/2 ,is a hamiltonian circuit

Ore's Theorem:

if G is a simple graph of n vertices ,where n>=3  such that deg(u)+deg(v)>=n for every pair of NONADJACENT vertices u,v in the graph..then G has a Hamiltonian CKt

Another theorem:

A vertices with no loop and paralle edges, which has atleast 1/2(n-1)(n-2)+2 edge ,is Hamiltonian

refer:http://mathonline.wikidot.com/dirac-s-and-ore-s-theorem

so i think all are true

4 Comments

@uddipto.. plz enlight about this reverse direction issue
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got it
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Question is given it is hamiltonian graph, your theorems are given this it is hamiltonian. So all hamiltonian graph need not satisfy this need not
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1 vote
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All of the Above, this is a duplicate question

Check here for explanation https://gateoverflow.in/115602/ugcnet-dec2016-ii-5

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1 is true: If G = (V, E) has n ≥ 3 vertices and every vertex has degree ≥ n/2 then G has a Hamilton circuit. proved theorem in book
http://www.sas.upenn.edu/~htowsner/uconnprev/math340f13/Lecture%207%20(Sep%2019).pdf


2 is true. explained in comment.

3 is true: deg v + deg w ≥ n for every pair of distinct non-adjacent vertices v and w of G (*)

https://en.wikipedia.org/wiki/Ore's_theorem

therefore: all are ture

option 4

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      Hence, option (A) is correct.

  • Any graph with n vertices and at least $\frac{1}{2}\left ( n-1 \right )\left ( n-2 \right ) + 2$ edges must be hamiltonian.

https://www.quora.com/Let-G-be-a-simple-graph-with-n-vertices-and-1-2*-n-1-n-2-+2-edges-How-do-I-use-Ores-theorem-to-prove-that-G-is-Hamiltonian#  // (Good read of answer by "Amitabha Tripathi " Professor of Mathematics at IIT Delhi).

      Hence, option (B) is correct.

  • Let G be a (finite and simple) graph with n ≥ 3 vertices. Degree of a vertex v ( i.e deg v ) in G, i.e. the number of incident edges in G to v. Then, Ore's theorem states that if 

deg v + deg w ≥ n for every pair of distinct non-adjacent vertices v and w of G, then G is Hamiltonian.       

 https://en.wikipedia.org/wiki/Ore's_theorem

    Hence, option (C) is correct.

Therefore, all options are true. Hence, Option (D) is correct.


For more explanation and validation of the above options in the question one may refer to following module of NPTEL.

https://nptel.ac.in/courses/111106050/Module6.pdf

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