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Each and every purana and itihasa has some number of parvas /skandas. Further there are many chapters inside every skanda. Did Vyasa himself name each Skanda and each chapter or it is done by later commentators? Did he give verse numbers as well like we see in today works?

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Name of the eighteen Parvas and their divisions are mentioned in Mahabharata, Adi Parva, SECTION II. This whole chapter is dedicated to Parvas and their divisions, so I am quoting only relevant parts.

The high-souled Vyasa composed these hundred parvas of which the above is only an abridgement: having distributed them into eighteen, the son of Suta recited them consecutively in the forest of Naimisha as follows:

  1. Adi Parva

    'In the Adi parva are contained Paushya, Pauloma, Astika, Adivansavatara, Samva, the burning of the house of lac, the slaying of Hidimba, the destruction of the Asura Vaka, Chitraratha, the Swayamvara of Draupadi, her marriage after the overthrow of rivals in war, the arrival of Vidura, the restoration, Arjuna's exile, the abduction of Subhadra, the gift and receipt of the marriage dower, the burning of the Khandava forest, and the meeting with (the Asura-architect) Maya.

  2. Sabha Parva

    The second is the extensive parva called Sabha or the assembly, full of matter.

  3. Aranyaka or Vana Parva

    Then comes the third parva called Aranyaka (relating to the forest)

  4. Virata Parva

    "The extensive Parva that comes next is called Virata.

  5. Udyoga Parva

    "Listen then to (the contents of) the fifth Parva which must be known as Udyoga.

  6. Bhishma Parva

    "Then is recited the Bhishma Parva replete with wonderful incidents.

  7. Drona Parva

    "Then is recited the wonderful Parva called Drona full of incidents.

  8. Karna Parva

    "Then comes the most wonderful Parva called Karna.

  9. Shalya Parva

    "Then hath been recited the wonderful Parva called Salya.

  10. Sauptika Parva

    "Then shall I describe the Parva called Sauptika of frightful incidents.

  11. Stri Parva

    "After this hath been recited the highly pathetic Parva called Stri,

  12. Shanti Parva

    "Twelfth in number cometh the Santi Parva,

  13. Anushasana Parva

    "Next in order is the excellent Anusasana Parva.

  14. Ashwamedhika Parva

    "Then comes the fourteenth Parva Aswamedhika.

  15. Ashramavasika Parva

    "Then comes the fifteenth Parva called Asramvasika.

  16. Mausala Parva

    "After this, you know, comes the Maushala of painful incidents.

  17. Mahaprasthanika Parva

    "The next is Mahaprasthanika, the seventeenth Parva.

    "In this, those foremost among men the Pandavas abdicating their kingdom went with Draupadi on their great journey called Mahaprasthana.

  18. Svargarohanika Parva

    "The Parva that comes after this, you must know, is the extraordinary one called Svarga of celestial incidents.

It also consists Harivamsa.

"The above are the contents of the Eighteen Parvas. In the appendix (Khita) are the Harivansa and the Vavishya. The number of slokas contained in the Harivansa is twelve thousand."

In Shiva Purana, Section 1 - Vidyeśvara-saṃhitā, Chapter 2, name of all the Samhitas are mentioned as follows:

49-51. Originally the Śivapurāṇa was of very enormous size consisting of twelve sacred Saṃhitās:—(1) Vidyeśvara (2) Rudra, (3) Vaināyaka, (4) Aumika, (5) Mātṛ (6) Rudraikādaśa, (7) Kailāsa, (8) Śatarudraka, (9) Sahasrakoṭirudra, (10) Koṭirudra, (11) Vāyavīya and (12) Dharmasaṃjña. O brahmins, I shall mention the number of verses in those Saṃhitās. Please listen with due attention.

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It is highly unlikely that a single human being could have authored so many Purāṇas, divided them into chapters, numbered the verses and so on. For comparison, the Āndhra Mahābhārata was translated from Sanskrit to Telugu by 3 poets over a period of 3 centuries under 3 different kings.

This is what the Viṣṇu Purāṇa says on how it came about:

Accomplished in the purport of the Purāṇas, Vyāsa compiled a Paurāṇik Saṃhitā, consisting of historical and legendary traditions, prayers and hymns, and sacred chronology. He had a distinguished disciple, Sūta, also termed Romaharṣaṇa, and to him the great Muni communicated the Purāṇas. Sūta had six scholars, Sumati, Agnivarchas, Mitrayu, Śānśapāyana, Akritavraṇa, who is also called Kāśyapa, and Sāverṇi. The three last composed three fundamental Saṃhitās; and Romaharṣaṇa himself compiled a fourth, called Romaharṣaṇika. The substance of which four Saṃhitās is collected into this (Viṣṇu) Purāṇa.

So it's more plausible that Vyāsa's (Kṛṣṇa Dvaipāyana) disciples authored and then dedicated all their works to their guru.

Swami Harshananda says the same in A Concise Encyclopaedia Of Hinduism (Vol. 2):

Evolution of the Purāṇas

Though it is conceded that the purāṇas are of hoary antiquity, extreme paucity of information leaves us in complete darkness with regard to their character or contents, none of which seems to have come down to us in the original form. No doubt Hindu tradition attributes the authorship of these purāṇas – the eighteen Mahāpurāṇas (the main purāṇas) and the eighteen Upapurāṇas (the subsidiary purāṇas) – to the sage Kṛṣṇa Dvaipāyana, better known as Vedavyāsa or Vyāsa.

However, this cannot be substantiated by the evidence available. The original purāṇa referred to in the Vedic and allied literature was, perhaps, a conglomeration of ākhyānas (tales), upākhyānas (anecdotes), gāthās (metrical songs or proverbial sayings current in the ancient society) and Kalpakoṭis (sayings that had come down through the ages). The sage Vedavyāsa might have compiled these into one Purāṇasaṃhitā. His disciples and their disciples as also others in that tradition might have composed more detailed works which gradually took the present form, the eighteen purāṇas as we know them today. This surmise is confirmed by the accounts given in some of the more ancient purāṇas like the Vāyupurāṇa, the Brahmāṇḍapurāṇa and the Viṣṇupurāṇa. According to them, after compiling the original Purāṇasaṃhitā, Vyāsa imparted it to his disciple Sūta Romaharṣaṇa (also spelt as Lomaharṣaṇa), who in his turn made it into six versions and taught them to his six disciples. Of these, three disciples viz., Kāśyapa, Sāvarṇi and Śāṃsapāyana made three separate saṃhitās which were named after them. These three, along with that of Romaharṣaṇa are known as 'mūlasaṃhitās'. The later purāṇas were evolved out of these.

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