Determine the minimum number of separate racks needed to store the chemicals given in the table (1st column) by considering their incompatibility u sing graph coloring technique. Clearly state you steps and graphs used. Chemical: Incompatible with Ammonia (anhydrous): Mercury, chlorine, calcium hypochlorite, iodine, bromine, hydrofluoric acid (anhydrous) Chlorine: Ammonia, acetylene, butadiene, butane, methane, propane, hydrogen, sodium carbide, benzene, finely divided metals, turpentine Iodine: Acetylene, ammonia (aqueous or anhydrous), hydrogen Silver: Acetylene, oxalic acid, tartaric acid, ammonium compounds, pulmonic acid Iodine: Acetylene, ammonia (aqueous or anhydrous), hydrogen Mercury: Acetylene, pulmonic acid, ammonia Fluorine: All other chemicals
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Step 1. We assign chemical 1 (Ammonia), the first vertex and the first color (blue).
Step 2. Now we analyze the compatibility with chemical 2 (Chlorine). We note that they are incompatible, therefore, chemical 2 must be placed in another rack (placed at vertex 2, red color).
Step 3. We evaluate the compatibility of chemical 3 (Iodine) with chemical 1 (Ammonia), and we see that they are incompatible. Now we analyze the compatibility of chemical 3 (iodine), with chemical 2 (chlorine), and we see that both are compatible, therefore, they can be placed in the same rack (chemical 3 is placed, at vertex 2, color Red).
Step 4. We evaluate the compatibility of chemical 4 (Silver), with chemical 1 (Ammonia), we see that both are compatible, therefore, they can be placed in the same rack (chemical 4 is placed, at vertex 1, color blue).
Note: Note that Silver is also compatible with Chlorine and Iodine, therefore, it can also be located at vertex 2 (red color), without affecting the result.
Step 5. We evaluate the compatibility of chemical 5 (Iodine). In this case, we noticed that we had previously labeled this chemical (chemical 3), and it was placed in vertex 2, red.
Step 6. We evaluate the compatibility of the chemical 6 (Mercury), We check that it is incompatible with the chemical 1 (Ammonia), but it is compatible with the chemicals located in vertex 2, therefore, it is placed in the vertex 2, red color.
Step 7. We evaluate the compatibility of chemical 7 (Fluorine), we see that it is incompatible with all other chemicals, therefore, we place it in a new vertex: vertex 3, green color.
In this way we get the answer: a minimum of 3 separate racks to store the chemicals are needed. This is the minimum number of colors, that are needed to color the vertices of the graph, and is known as a chromatic number.