As I understand, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. Should I show, that chromatic numbers of all Petersen's subgraphs are not equal to it clique number? Maybe there is an easier and quickly way to show that.
Why is the Petersen graph not a perfect graph?
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graph-theory
2 Answers
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The Petersen graph is a Kneser graph, precisely $\text{KG}(5,2)$. By Lovasz theorem the chromatic number of $\text{KG}(n,k)$ is $n-2k+2$, hence in our case $\chi(G)=3$. On the other hand $5<6$, hence the Petersen graph is triangle-free.
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0Is there any theorem, that perfect graph can't be triangle-free? – 2017-02-01
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0Using the Lovasz theorem on Kneser graphs seems like major overkill. The Peterson graph contains an induced pentagon, qed. – 2017-02-01
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0@Nate: I agree with you. I was just in the mood of killing mosquitoes with atomic bombs. – 2017-02-01
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Check out this link --- it shows the clique number is less than the chromatic number of the entire graph.
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0Thanks, I saw it. But, I didn't clearly understand where the proof, that the clique number is less – 2017-02-01