We will start our analysis of this topic with the first basic statement, as per the epic, Drupada always desired for his daughter Draupadi to be married to Arjuna, there was never any question in the mind of Drupada who his son-in-law would be. So, it could be possible for Draupadi to grow up hearing the tales of Arjuna's valour and might have been attracted to him initially.

1. Swayamvar Contest, initial reaction of Draupadi and aftermath at Potter's house [BORI CE Volume 1 – Section Twelve – Droupadi-Svayamvara Parva]: On seeing the target shot and on seeing Partha (Arjuna), the equal of Shakra, Draupadi was extremely happy. She went to Arjuna with a garland of white flowers. He soon left the arena, followed by the one who would be his wife.

Below is in the same parva, Dhristadyumna describes how Draupadi left with Arjuna from the arena:

Like an elephant bride happily follows the male elephant, Draupadi followed, holding the deerskin of Arjuna.

We can see the first sparks of loving relation here, even though Arjuna was disguised as a Brahmana, but that did not matter to Draupadi and she was extremely happy to see that youth hit the target. Draupadi in her mind must have already made up that Arjuna as her husband after target was pierced and she garlanded him. So, is it not natural for Draupadi to love Arjuna a bit more as he was the one who won her and she was extremely happy to see him win her?

Events at the potter's house: Draupadi was not asked for her opinion with regards to the marriage with the five Pandavas, so we are not sure what would have been her answer. Nevertheless, there is no record of her protest, so we can be fairly sure that she did not object to it, but happily accepting the proposition is NOT recorded as well, as she was not asked for her opinion in the first place. She simply did her duty cheerfully of arranging dinner for the Pandavas on the instructions of Kunti.

So, with the above, we are clear that on two instances, extreme happiness has been recorded with regards to Draupadi on account of Arjuna winning her.

2. Arjuna returns back to Khandavaprastha after his 12 year exile [BORI CE – Volume 2 - Section Eighteen]:

Below is the passage, wherein Arjuna returns with Subhadra after his exile period was over.

Arjuna went to Draupadi, then, out of love, Draupadi told Arjuna 'O Kunti's son! Go to the daughter of the Satvatas. A second load always loosens the first tie, however strong'. Draupadi thus lamented in many ways. Arjuna pacified her a lot and asked for forgiveness. He quickly went to Subhadra, who was dressed in red silk. Arjuna sent her away to dress in garments worn by a cowherd lady.

As far as my reading of the epic, I did not find any such lamentation on part of Draupadi when any of her other husbands remarried. Interestingly, none of her other husbands apart from Arjuna asked for her forgiveness when they remarried, which implies that Draupadi was not really bothered when other Pandavas had remarried, so none of them had to ask for her forgiveness. In my view, this incident highlights the special relationship shared by Arjuna/Draupadi.

3. Does Draupadi mentioning the names of the Pandavas in the chronological order signify her preference? No, it doesn't show any preference but she usually mentions from her eldest husband to the youngest husband.

We can refer to the instances mentioned below to decipher the above in more detail:

A. When Draupadi was asked by Dhritarashtra for boons after the incidents in the Dyuta Sabha, she had asked for the two boons [BORI CE – Volume 2 – Section Twenty Seven – Dyuta Parva]

a. First boon was to have Yudhisthira be freed from slavery;

b. Second boon was to desire Bhima, Dhananjaya, Nakula and Sahadeva not be slaves.

We see the chronology from the eldest (Yudhisthira) to the youngest (Sahadeva) has been maintained by Draupadi.

B. When Draupadi complains to Lord Krishna about the events which happened during Dyut Sabha [BORI CE – Volume 2 – Section Thirty-One – Kairata Parva]

Shame on Bhimasena's strength and shame on Partha's Gandiva. Both of them allowed me to be oppressed by insignificant ones.

We see the chronology from the eldest (Bheema) to the youngest (Arjuna) has been maintained by Draupadi.

C. When Jayadratha was taking away Draupadi forcibly in his chariot, he sees five chariots racing towards him and asks Draupadi to point to him each of the Pandavas riding those chariots [BORI CE – Volume 3 – Section Forty Two – Droupadhi Harana Parva]

Draupadi again mentions about each of her husbands in chronological order from the oldest to the youngest, below is what she specifically says about Bhima and Arjuna;

For Bhima: You will see a mighty-armed one seated on his chariot, like a fully grown shala tree. He is biting his lips and his brows are furrowed. His name is Vrikodara and he is my husband. He is immensely strong and brave and is drawn by well-controlled and strong horses. The deeds that he has done are superhuman. He is known on earth by the name of Bhima. Nothing is left of those who commit crimes against him. He never forgets an enemy. He thinks of every possible way to avenge the enmity and even after that, he is not pacified.

For Arjuna: He is gentle, generous, forbearing and famous. He is brave among men. He is in control of his senses and serves his elders. He is Yudhisthira's brother and disciple. His name is Dhananjaya and he is my husband. Desire, fear or avarice will never cause him to abandon dharma. He never performs a cruel act. His energy is equal to that of the fire-god. He is Kunti's son and he withstands and represses all his enemies. He is knowledgeable about everything concerning dharma and artha. The learned one removes the fear from those who are frightened.

As you can see from the above description, Draupadi was proud of Bhima's strength and bravery. But, the same Draupadi seems to have been even more proud of Arjuna, as she adds much more to Arjuna. Arjuna was a refined and composed character, who was not only extremely brave, charismatic, but also gentle in manners and generous in nature and at the same time was a learned one and in control of his senses.

Basically, in Draupadi's eyes, Arjuna had a combination of various qualities of all the Pandavas, like knowledge of dharma, gentleness, generosity, learning, bravery, sense of control. When someone possess all these desirable qualities, is it not natural to have a little bit of more love towards that person as compared to the others? For example, if a student tops the class consistently with his hard work/intelligence and shows respect to the teachers all the time, then naturally the teachers will show more love to that pupil, right?

In the same parva, below is what Draupadi separately mentions to Jayadratha prior to the above passage, here she explicitly mentions much more on Arjuna compared to Bheema and here she does not follow chronology and in fact starts with Arjuna first:

For Arjuna:

Both Krishnas (Lord Krishna/Arjuna) will follow my footsteps, riding together on a single chariot. Indra himself cannot abduct me, not to speak of a weak and wretched human. Kiriti (Arjuna), the slayer of enemy heroes, terrifies the hearts of enemies on his chariot. When he enters your army for my sake, he will consume it like the summer's dry wood. The arrows released from Gandiva by Dhananjaya's arms are extremely swift and make a thunderous noise. They will make a terrible roar. The large showers of arrows released from Gandiva are extremely swift and are like locusts. There will be the roar of conch shells and the roar of leather gloves. The wielder of the Gandiva will release them repeatedly. When those arrows pierce your chest, what thoughts will pass through mind?

For Bhima:

You will see Bhima advance, with a mace in his hand.

Draupadi mentions a whole paragraph about Arjuna to Jayadratha but a single sentence on Bheema. This is not to say that Bheema was any weaker in strength to Arjuna but Draupadi seems to be fond of explaining much more in detail when it came to Arjuna.

So, with the above instances we have seen, Draupadi usually followed to mention her husbands from the eldest to the youngest and on a few occasions this was not followed, like the case above where Arjuna's bravery is mentioned first and in much more detail as compared to Bheema.

4. BORI- Volume – 3 – Section Thirty Three – Tirtha Parva:

Below is the passage which describes how Draupadi asks Bhima to fetch her flowers

"O Bhima! Look at this divine, dazzling and supreme blossom. It is full of fragrance and has delighted my heart. O destroyer of enemies! I will give this to Dharmaraja. But to satisfy my desire, get me others, so that I can take them back to the hermitage in Kamyaka. O Bheema! If I am your beloved, get them for me in large numbers. I wish to take them back to the hermitage in Kamyaka".

5. BORI – Volume – 4 – Section Forty Six – Kichaka Vadha Parva

Below are the excerpts after the harassment of Draupadi by Kichika.

"What will I do? Where will I go? How will my task be accomplished?" While she was thinking in this way, Bhima surfaced in her mind. "Other than Bhima, no one can act so as to bring pleasure to my mind"

"O Bhimasena! Arise! Arise! How can you sleep as if you are dead? The evil one, who molested the wife of someone who is not dead, is still alive. My enemy, the evil-doing general, is still alive."

On being woken up, below is what Bheema says to Draupadi.

"Why have you come to me in this hurried fashion? Your complexion is not natural. You appear to be pale and wan. Tell me everything in detail, whether it is pleasant or unpleasant, agreeable or disagreeable, so that I know. Having heard everything, I will decide what should be done next. O Krishna! I am trusted by you in all deeds. I have repeatedly saved you from all dangers. Quickly tell what you desire, what task needs to be done. Then return to your bed before the others arise."

6. BORI – Volume – 4 – Section Forty Six – Kichaka Vadha Parva

Below is what Draupadi says how she feels about the miserable state of Bheema and Arjuna in the Virata kingdom:

For Bheema: I am bereft of my senses when you fight with tigers, buffaloes and lions in the inner quarters and Kaikeyi watches you. On seeing me unconscious, Kaikeyi will arise and tell her women, 'I think this affection is born out of this sweet-smiling one living with the cook and she sorrows when he fights with these immensely brave beings. Sairandhri is beautiful in form and Ballava is extremely handsome. The minds of women are impossible to fathom. But it seems to me that they are made for each other. Sairandhri is always overcome by pity because they happily live together. They have lived in this royal household for the same duration of time.'. Through such words, she always makes me known. When she sees me angered, she suspects that I attached to you. When she utters such words, I am over taken by great grief.

In about half of the above paragraph, it was Kaikeyi (Queen Sudeshna) who was describing about Bheema and Draupadi's relationship.

For Arjuna:

"Alone on a single chariot, he defeated gods, men and serpents. That youthful one is now a dancing master for King Virata's daughter. The one with the infinite soul satisfied the fire god in Khandava. Partha has now gone to the inner quarters, like a fire covered in a well. He was a bull among men and enemies were always frightened of him. Dhananjaya is now in a form that is despised by the world. Enemies trembled at the twang of his bow and the slapping of his palms. He now pleases women with sweet sounds of his singing. A diadem that was like the sun always adorned his head. Dhananjaya's unkempt hair is now braided. All the celestial weapons are known to that great-souled one. He is the repository of all knowledge and wears earrings. Thousands of kings, whose energy was unlimited, could not cross him and overcome him in battle, just as the great ocean does not cross the shoreline. That youthful one is now a dancing master for King Virata's daughter. O Bhima! The earth, with its mountains and forests and mobile and immobile objects, trembled at the roar of his chariot. He is the immensely fortunate one whose birth destroyed Kunti's sorrow. O Bhimasena! I now sorrow over your younger brother. He is adorned in golden ornaments and earrings and sports conch shells in his hands. On seeing him approach, my mind is immersed in sorrow. O Bhima! The archer Dhananjaya wears his unkempt hair in a braid. O Bhima! On seeing him surrounded by young maidens, my mind is immersed in sorrow. He is equal to a God. When I see Partha surrounded by young maidens, in the midst of musical instruments, like a bull-elephant in rut surrounded by she-elephants, and in the service of Virata, king of Matsyas, who pays him, I can no longer see any directions. Surely the arya (Kunti) does not know the difficulties Dhananjaya has to confront"

We can clearly see how distressed was Draupadi with regards to Arjuna's state, sure she was distressed with every Pandavas' state of misery during their exile but the repeated sorrowful statements and at great length is reserved for Arjuna. Here too, Draupadi basically refers to Arjuna as an all-round dynamic personality apart from his bravery. Draupadi refers to Arjuna multiple times as a great-souled one, youthful one, and expresses her sorrow multiple times. She refers to him as one who is equal to a God. Also, she seems to have been distressed to see young maidens surrounding him and quotes it multiple times, she shows her possessiveness towards Arjuna I guess :)

7. BORI-CE, Book: The Mahabharat Volume 2 (Section Thirty-Two),

Below is the passage, wherein Draupadi feels disheartened due to the absence of Arjuna, who first goes to Himalayas to do tapasya to get blessings from Lord Shiva and then goes to heaven to acquire divine weapons.

After Dhananjaya's departure, those bulls among men lived there, unhappy in their minds and anxious about him. 'In particular, Panchali remembered the husband who was in the middle, the brave one who was absent. She told the foremost of the Pandavas, "in the absence of the two-armed Arjuna, the equal of the many-armed Arjuna (Kartavirya Arjuna) and the best of the Pandavas, this forest seems cheerless to me. Wherever I look, the earth seems to be empty to me. This forest, with its many marvels and blossoming trees, no longer seems to be attractive in Savyasachi's absence. The twang of his bow is like the roar of the thunder. I remember Savyasachi and without him, I cannot find any peace of mind"

As per my understanding, even though Draupadi was vocal and when needed spoke in public too on the matters related to Dharma, responsibilities, duties and so on but she did not usually show her feelings or her equation in public as with regards to her husbands. Only Arjuna seems to be an exception to such norms, as usually Draupadi was not very discreet or failed to be discreet with regards to her feelings towards Arjuna.

8. BORI-CE, Book: The Mahabharat Volume 10 (Section Eighty-Nine),

Below is the passage wherein Yudhisthira wants to know the reason why Arjuna is always on the move and has borne a disproportionate share of sorrow. Yudhisthir goes on to say that he doesn't see any inauspicious things on Arjuna's body that can explain it, so wants to know from Lord Krishna the reason.

'Having been thus addressed, Hrishikesha reflected for a very long time. He replied 'O king! No inauspicious marks can be seen in him (Arjuna), but for the fact that the lion among men has cheekbones that are exceedingly large.' Thus addressed by Lord Krishna, Yudhisthir told Lord Krishna that this was indeed true. However, Draupadi cast a sidelong glance towards Krishna (the sidelong glance signified disapproval).

This to me is a very amusing episode indeed, no less than a filmy scene, where it looks like Lord Krishna was in a playful mood and seems to pull the leg of Draupadi. Again, I don't find any other incident in the epic resembling anything remotely similar to this. This to me again depicts how Lord Krishna saw Arjuna and Draupadi and their relationship, throughout the epic, he treated them as his friends and they were like a best friends group.

9: BORI CE, Book: The Mahabharat  Volume 3 - Section Thirty-Five - YAKSHA-YUDDHA PARVA

Below is the passage wherein Arjuna after returning from the heaven presents gifts to Draupadi.

"When the chariot of the supreme god had departed, Shakra's great-souled son (Arjuna), the destroyer of all enemies, presented all the expensive and beautiful gifts that Shakra had given. The ornaments that he (Arjuna) lovingly gave to his beloved, Sutasoma's mother (Draupadi), glittered like the sun's rays. Then he seated himself in the midst of the bulls of the Kuru lineage, resplendent like the sun and the fire."

10: BORI CE, Book: The Mahabharat Volume 4 - Section Forty-Five - VAIRATA PARVA

Below is the passage wherein Pandavas along with Draupadi were marching towards Virata kingdom to spend their time in incognito. 

"Having reached that country, Draupadi told the king, "Behold. Many footpaths and different kinds of fields can be seen here. It is clear that Viarata's capital is still far away. My fatigue is extremely great. Let us spend the night here." Yudhishthira replied, "O Dhananjaya (Arjuna)! O descendant of the Bharata lineage! Lift Panchali and carry her. Once we free ourselves from this forest, we will settle down in the capital." Like a king of elephants, Arjuna swiftly raised Draupadi up. Arjuna put her down when they reached the outskirts of the capital."

11. BORI-CE, Book: The Mahabharat Volume 10 (Section Ninety-Four),

Below is the passage wherein Yudhisthir explains to Bheema on the reason of Draupadi's fall.

'She had a great partiality for Dhananjaya. She has reaped the fruits of that'

Please note, I don't agree to Yudhisthira's line of thinking here with regards to Draupadi's fall, she might have had a partiality to Arjuna but I don't think it was the reason for the fall. It was just Yudhisthira's personal opinion just like his opinion of fall for others.

As per my understanding, Draupadi died first, so she could die as a married woman instead of a widow. And, also the other Pandavas had fallen based on their age, like Sahadeva, Nakul, Arjuna and Bheema. I believe this episode was an illusion created by Gods to know the steadfast nature of Yudhisthira towards Dharma and Yudhisthira wins this test by the power of his Dharma.

Just like his opinion on Draupadi, Yudhisthira uses some statements which could be termed more of his personal opinions. He says Bheema had fallen just because he enjoyed eating more. There is no recorded instance in the epic which states that Bheema made others deprived of food because of his appetite, yes he was a voracious eater but he never made his loved ones go hungry and without food just because of his voracious appetite.

Yudhisthira says the reason that Arjuna has fallen is because Arjuna said that he could destroy Kaurava army in one day but failed to do so. In fact, Arjuna said that the divine astra which he was blessed with by Lord Shiva, he could annihilate Kauravas in the twinkling of the eye but he would not resort to using such powerful weapon as it could cause doom for the world.

Yudhisthira says Sahadeva was proud of his intelligence, but there is NO single instance in the epic which I have read so far which shows Sahadeva as a haughty person due to his intelligence. He was always a composed and humble individual even though he was the wisest person amongst the Pandavas.

So, we could attribute the reasons which Yudhisthira has given to be more of personal opinions.

As a said earlier, this is just a small exercise to add all the instances of their relationship and add our viewpoints based on the instances available to us.