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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
in Graph Theory by Junior | 71 views
1 seems the most suitable answer
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?
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
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
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
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!
C4 is a 2 regular graph and cycle contain then how 1 statement is false?

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