Top MCQs on Graph Theory in Mathematics

Question 11

Which one of the following is TRUE for any simple connected undirected graph with more than 2 vertices?

• No two vertices have the same degree.

• At least two vertices have the same degree.

• At least three vertices have the same degree.

• All vertices have the same degree.

Question 12

The chromatic number of the following graph is _________ . Note - This was Numerical Type question.

• 2
• 4
• 3
• 5

Question 13

Which of the following statements is true for every planar graph on n vertices?

• The graph has a vertex-cover of size at most 3n/4
  

• The graph is Eulerian
  

• The graph is connected
  

• The graph has an independent set of size at least n/3
  

Question 14

G is a graph on n vertices and 2n - 2 edges. The edges of G can be partitioned into two edge-disjoint spanning trees. Which of the following is NOT true for G?

• There are two edge-disjoint paths between every pair to vertices
  

• The minimum cut in G has at least two edges
  

• For every subset of k vertices, the induced subgraph has at most 2k-2 edges
  

• There are two vertex-disjoint paths between every pair of vertices
  

Question 15

Let G be the non-planar graph with the minimum possible number of edges. Then G has

• 10 edges and 6 vertices
  

• 10 edges and 5 vertices 
  

• 9 edges and 6 vertices
  

• 9 edges and 5 vertices 
  

Question 16

Which of the following graphs has an Eulerian circuit? 

• The complement of a cycle on 25 vertices 
  

• A complete graph on 90 vertices
  

• Any k-regular graph where kis an even number.
  

• None of the above 
  

Question 17

Let G=(V,E) be a directed graph where V is the set of vertices and E the set of edges. Then which one of the following graphs has the same strongly connected components as G ? 

• B
  

• A
  

• C
  

• D
  

Question 18

Consider an undirected graph G where self-loops are not allowed. The vertex set of G is {(i, j): 1 <= i <= 12, 1 <= j <= 12}. There is an edge between (a, b) and (c, d) if |a − c| <= 1 and |b − d| <= 1. The number of edges in this graph is __________.

• 500

• 502

• 506

• 510

Question 19

An ordered n-tuple (d1, d2, … , dn) with d1 >= d2 >= ⋯ >= dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1, d2, … , dn respectively. Which of the following 6-tuples is NOT graphic?

• (1, 1, 1, 1, 1, 1)
• (2, 2, 2, 2, 2, 2)
• (3, 3, 3, 1, 0, 0)
• (3, 2, 1, 1, 1, 0)

Question 20

The maximum number of edges in a bipartite graph on 12 vertices is __________________________.

• 36
• 48
• 12
• 24