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@Yespat. I see you are located in 7a so you don't have a lot of time left for them to mature.
The vines through the leaves contribute significantly to the plant's well being and thus I would NOT recommend removing vines. It will not speed up the ripening process and very likely will detract significantly from the health of the plant.
You can increase the size of some melons by pruning smaller or less healthy individual melons. This allows the plant to focus its energy on developing fewer but larger and healthier fruits. However, this practice will not speed up ripening and in fact seems likely to delay it while the individual melons remaining grow larger.
Depending on the variety you selected, the days to maturity can...