In 1920, an M L Collins (36) was in Shreveport with wife Cora and son M L Jr.

In 1900, Milton and Malissa [Thompson] were in Caddo parish with their 9 living children (of 11 born): Willie, Turner(?), Melaile [Melvin?], Susie?, Lenora, Erie, Jennie and Mister?(son).

Nearby is an 'Edmond' Collins with wife Nancy [Ramble?] and two children.

In 1880, Milton and Malissa are in the 8th ward (Caddo) with children Edmond, Willie, and Tenn? [Turner?]. Milton was born in AL; both parents were born in GA. Malissa was born in LA.

Nearby is an 'Edmond' Collins (37) with with wife Nancy. Both were born in AL. Their fathers were both born in GA. Their children's names were Sarah L., Harkles, Licey/Lucey?, Mary and Sallie.

Also in the neighborhood was Miles Collins (29) and wife Lazina. Their children were Emma, Henry and Joshua.

Enumerated next to Miles and Milton was Harkless (57) from GA. In the household was son Green (24) and g-dau Julia. Harkless was born in GA. His parents were born in VA (father) and Tenn (mother).

In 1870, Harkins Collins (50, b. AL) and wife Lili (40, b. AL) were with children Miles (19, b. AL), Green (17, b. AL), MILTON (14, b. AL) and Maggie (12, b. LA).

Next to Harkins was Grace (Green?, 45, b. AL) and Esther [Sims] with their children Henrietta, Dora, Clark, Jerry, Leah and Purata (?).

Then, there was Matthew(21) with Ida (17), Green B., and "Dibbee" Daniel (64). Dibbee/Debbie? (female) was born in NC.

So now you have what appears to be Melvin's immediate family, his father, siblings, grand-father, and grand aunt and uncles. Of course, further proof is needed. The next step would be to look for marriage and death records. Marriage records are online for Caddo Parish (see link below).

I would start with Melvin's death record. Do you know when and where Melvin died? Any death record could have the name of his parents, including his mother's maiden name (verification). Of course, any of his sibling's death record could also have the same information.
What about his children? Their death record could have the maiden name of their mother.

Work from known ancestors back in time.
Perhaps someone on the board will find Melvin and/or Cora in a state death index.
There are people by both names in the AR death index, but it would be good to have further evidence of their migration to Arkansas before assuming these are your 'Melvin' and 'Cora'.