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Ancient DNA has been used to show aspects of Neanderthal appearance. A fragment of the gene for the melanocortin 1 receptor (MRC1) was sequenced using DNA from two Neanderthal specimens from Spain and Italy, El Sidrón 1252 and Monte Lessini (Lalueza-Fox et al. 2007). Neanderthals had a mutation in this receptor gene that has not been found in modern humans. The mutation changes an amino acid, making the resulting protein less efficient. Modern humans have other MCR1 variants that are also less active resulting in red hair and pale skin. The less active Neanderthal mutation probably also resulted in red hair and pale skin, as in modern humans.
The specific MCR1 mutation in Neanderthals has not found in modern humans (or occurs extremely rarely in modern humans). This indicates that the two mutations for red hair and pale skin occurred independently and does not support the idea of gene flow between Neanderthals and modern humans. Pale skin may have been advantageous to Neanderthals living in Europe because of the ability to synthesize vitamin D.
... in other words, red hair in modern humans (very likely) wasn't inherited from neandertals.