The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
+7 votes

For an undirected graph $G=(V, E)$, the line graph $G'=(V', E')$ is obtained by replacing each edge in $E$ by a vertex, and adding an edge between two vertices in $V'$ if the corresponding edges in $G$ are incident on the same vertex. Which of the following is TRUE of line graphs?

  1. the line graph for a complete graph is complete
  2. the line graph for a connected graph is connected
  3. the line graph for a bipartite graph is bipartite 
  4. the maximum degree of any vertex in the line graph is at most the maximum degree in the original graph
  5. each vertex in the line graph has degree one or two
asked in Graph Theory by Veteran (111k points)
retagged by | 427 views
Option B). is true.
yes, Agreed. (y)
why not option C?
you can check with K(2,2). The line graph obtained is not biparitite.

2 Answers

+8 votes

option B is the right answer

We can solve this question by eliminating options

Option (A) False

Let us take a complete graph of  $4$ vertices:

In a line graph, no. of edges is

$\sum_{i=0}^{n}$  ${}^{d_{i}}C_{{2}},$     $d_{i}$is degree of each vertex

=$\frac{3\times 2}{2}+\frac{3\times 2}{2}+\frac{3\times 2}{2}+\frac{3\times 2}{2} = 3\times 4=12$

No. of vertices in line graph = No. of edges in original graph
No. of vertices in line graph $= 6$
So, no. of edges to make complete graph with $6$ vertices $= \frac{6\times 5}{2} = 3\times 5=15$
But for given line graph from complete graph of $4$ vertices we have only $12$ edges.

Option (B)   True

1. Smallest line graph for original graph one edge

     which is also connected graph

If a graph is connected with more then one edge, it will never be disconnected 

Option (C)    False

This cannot be 2-colorable and hence is not bipartite. 

Option (D)    False

Because line graph degree of vertex depends on the attribute

e.g., $\left [ A,B \right ]$ as point in line graph, then degree of this vertex depends on degree of $A$ and degree of $B$ in the original graph.

I'm drawing degree for a point $[AB]$ in line graph.

So, this is wrong.

Option (E)    False (wrong as proved in above option (D))

answered by Active (4.9k points)
edited by
+2 votes

The line graph of a connected graph is connected. If G is connected, it contains a path connecting any two of its edges, which translates into a path in L(G) containing any two of the vertices of L(G). Therefore, option B is correct.

We can also do this question using elimination of options. 

You can view the following google drive link for the example -

answered by Active (1.7k points)
edited by
but your L(G2): e1---e2 it is bipertite right?e1 may be in one partition and e2 on is bipertite i guess.

I have corrected the example, and updated the file.
great example you have updated...proving line of tree is not a tree too :p

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

44,157 questions
49,640 answers
65,808 users