# Understanding connection between independent set and chromatic number

I have came across following facts / definitions:

1. Maximum independent set: Independent set of largest possible size.
2. Maximal independent set: Independent set such that adding any other vertex to the set forces the set to contain an edge
3. Independence number: Number of vertices in the maximum independent set
4. Relation between chromatic number ($$χ$$) and independence number ($$β$$) We can color vertices in maximum independent set themselves in single color and they will form largest number of vertices with same color. Hence, largest number of vertices with same color cannot exceed the independence number. Hence, $$β≥\frac{n}{χ}$$.
5. Relation with chromatic number: Chromatic number = size of minimum maximal independent set
6. Chromatic partitioning involves partitioning all vertices of graph into the smallest possible number of "disjoint" independent sets.

My doubts:

D1. Is fact 5 simply wrong?
D2. Is that "disjoint" in fact 6 should really be "minimal"?
D3. Is every maximum independent set always maximal?

D1 : this is wrong. For example the complete bipartite &K_{5,5}& has two maximal independent set, each one of size 5. The chromatic number is 2, not 5. I think it should be something close to $$n$$ divided by the minimal maximal independent set size (to check), using something close to your previous definition.
D3 : no. Consider the complete bipartite graph $$K_{5,3}$$. The maximum independent set has size 5, but the other set of 3 vertices is also maximal (you cannot add a vertice and keep it independent)