Answer:  

Dr. Jiro Komiyama, who is actually studying dyeing clothes, kindly answers the question above.
 
 
Denim is twilled cotton cloth dyed with indigo. When oxygen reacts with indigo using the proper oxidant, oxygen is added to the double bond at the middle of the indigo molecule (see right figure: structural formula of indigo), causing that one indigo to become two benzene derivatives and the color disappears.
 
This type of reaction also occurs when a brownish cup or towel is whitened using bleach which includes oxidant. When cellulose, the polymer which makes up cotton thread, reacts with a strong oxidant, however, it becomes oxycellulose and the cloth becomes a rag.
 
So we choose oxidants which have different strengths, and use them under proper conditions (temperature, pH, concentration, etc), depending on how we want to bleach denim. For example, 

1. A partial break down of all the indigo dye contained in the complete cloth so that the color of the product becomes pale.
    

2. A complete break down of the indigo in part of the cloth so that this part will become white. 

We use sodium hypochlorite in the case of 1), and we use potassium permanganate in the case of 2). The reason why the clothes are exposed to sunshine after potassium permanganate process is to reduce the permanganate ion (purplish red color) to manganic acid (colorless) so that the bleaching reaction stops and the color of permanganate ion disappears.
  
Oxidants used in our daily life are:

1. Chlorine bleach including sodium hypochlorite
    

2. Bleach including sodium percarbonate which can be used for laundry

This B) type of oxidant does not break down the dye but only breaks down dirt (which is colored). So we use different types of oxidants appropriately, depending on our purpose.
 
 
Acknowledgement
We would like to thank Dr. Jiro Komiyama for his kind answer. We also would like to thank Dr. Tadashi Nakanishi for his help.
  
  

This article is translated by Chemistryquestion.com from the original article in Chemistryquestion.jp.  Please let us know if you find any errors.