I am studying graph theory with the textbook *introduction to graph theory* written by Douglas B. West. In the book, the **chromatic number** and **partite** are defined by ...

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Definition
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The **chromatic number** of a graph $G$ is the minimum number of colors needed to label the vertices so that adjacent vertices receive different colors.

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Definition
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A graph $G$ is **$k$-partite** if $V(G)$ can be expressed as the union of $k$ (possibly empty) independent sets, where $V(G)$ represents the set of vertices of $G$.

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Statement
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In addition to these two definitions, the author introduces a statement, "A graph is $k$-partite if and only if its chromatic number is at most $k$."

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My Question
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1. What does "at most" mean?
	* If the chromatic number can be $4, 5, 6$ or $7$, I can accept that $k$ is $7$. Given a graph, however, the chromatic number is a fixed integer. For example, the following graph's chromatic number is at most $3$? No, I think it is exact $3$.
	[![enter image description here](https://i.stack.imgur.com/vk5g8.png)](https://i.stack.imgur.com/vk5g8.png)
2. Without the phrase "at most", I agree with the necessity.
	* If a graph's chromatic number is $k$, it is $k$-partite ($\because$ $V(G)$ can be expressed as the union of $k$ independent sets.)
3. However, I do not agree with the sufficiency.
	* For any graph with $n$ verticies, its vertices can be expressed as the union of $n$ independent sets. (i.e., every vertices' color is different.) Then, the graph is $n$-partite. However, its chromatic number may be less than or equal to $n$. I think it may be $n$ when every vertices are adjacent pair-wisely. Thus, I think the statement that a graph is $k$-partite only if its chromatic number is at most $k$, is wrong!

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What is the problem of my thoughts? I believe that the author's claim is correct. Please answer without any slang. I can understand the literary English, but cannot the spoken English. Thank you for reading my question.