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Vertex Chromatic Numbers and Betti Numbers

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Prove for every graph G of order n, that n/B(G)<=X(G)<=n+1-B(G).

X(G) is the minimum integer k for which a graph G is k-colorable is called the vertex chromatic number

In the page 82 B(G) is defined like independent sets like you say but in the page 187 it other kind of B and it is define like Betti number and it is defined below

B(G) is the Betti number of the graph G of order n and size m having k components and it is defined as B(G)=m-n+k.

You can see on the page 187 of "Graph and Digraphs" 4 edition of G Chartrand and L. Lesniak.

Please can you explain what does B(G) mean and draw a graph.

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Solution Summary

Vertex Chromatic Numbers and Betti Numbers are investigated. The independent sets defined are examined.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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