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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 86 views
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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