I want to use graph 3-colorability to prove a problem is NP-complete But I'm not sure when a given graph is 3-colorable. I think if it doesn't have any node to be connected to all 3 vertices of a triangle in the graph.But i'm not sure. is it correct?
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No. Having a node connected to all three nodes in a triangle (which is just a more complicated way of saying "it has a clique of size 4") is sufficient but not necessary for the graph to not be 3-colorable. Graphs may be not 3-colorable for different reasons than having a clique of size 4.
For example:
It's not 3-colorable. Proof: the 5-cycle around the outside is not 2-colorable, and then you need an extra color for the middle vertex.
harold
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