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If a 2 regular graph G has a perfect matching, then which of the following is NOT true?
1. G is a cycle graph
2. Chromatic number of G is 2
3. Every component of G is even cycle
4. G is a bipartite graph
asked
Jan 16
in
Graph Theory
by
Chaitrasj
Junior
(
961
points)

59
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answer
comment
+1
1 seems the most suitable answer
0
Yes answer given is1st option
Why can't option 3 be answer? Why there will be a component of even cycle always?
Can you give some explanation like how you arrived at your answer?
+1
for perfect matching the necessary condition is the no of vertices should be even.
the graph being 2 regular implies the no of edges =no of vertices.
therefore the no of edges is even
having a graph with odd cycle violates the above conditions
0
Yes your explanation is right. But if every component of G is always an even cycle, means it's a cycle graph right... So how option 1 is becoming False under any certain case :(
What's the difference between option 1 and 3, I'm confused among them
+1
from my understanding a cycle graph consists of a single cycle.But our graph can contain multiple cycles and components.Hence 1 is the most suitable option
0
Yess we can have a graph with 2 components such that each component is $C_4$. In this case all other options are true, but 1 is becoming False it's not a cycle graph.. Thanks!
0
C4 is a 2 regular graph and cycle contain then how 1 statement is false?
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