Brown Pigment Around Tummy Tuck Scar 4 Weeks Post Op, Will It Go Away? (photo)

It's too early to predict the nature of your final scar, since you are only 1 week out from surgery.  Usually these early color changes fade with time, as the healing progresses and the inflammation settles. If, however, your pigment is stubborn - using some medical grade fade creams, like hydroquinone, will usually take care of these color changes.  Ask you PS if you need one, if the brown color is still present after a few more weeks.

Hyperpigmentation can make scarring more visible, and is worsened by sun exposure. You should avoid exposing your scar to the sun, particularly for the full year after your surgery. You can try silicone gel or silicone sheeting, which can help reduce the appearance of scars. Please ask your surgeon for their advice as well.

Your scar is in the early phases of healing and the photo is not concerning to me.  As most of the pigmentation is immediately under the tape I would ask if you had any itching or "rash" around the incision.  This could be an indication of a mild tape or adhesive allergy that has caused more inflammation in this area.

There's a great deal of information that goes on around an incision site right after tummy tuck surgery.  Most of it is manifested by a reddish color which indicates that the scar tissue that is forming is hyper vascular.  Over about a year this hyper vascular color fades as the scar matures.  Even if it does not fade in some occasions vascular lasers can help treat a red scar.
The brownish color that you notice may actually be more of an issue with hyper pigmentation.  Because you are fair skinned is likely that this will resolve on its own overtime but in hyper pigmentation certain creams or bleaching agents can be used to help diminish the amount of pigment.
I would be concerned that it's a little bit early to start with bleaching agents, but discuss your concerns with your plastic surgeon and he or she will guide you when the time is right.  Your plastic surgeon will also share with you his own regimen for helping to mature and soften the scars over time.  In my own practice I recommend massage with cocoa butter or vitamin E as a preliminary approach, but each surgeon will have his or her own specific regimen.
The timeframe on scar maturation is really over the course of a year so at four weeks out you are really quite early in the process and there is plenty of time for improvement.
Best wishes.

Your scar is going through the inflammatory phase of healing. It should go on to fade and you should monitor it for changes and report any redness or thickening in the scar to your plastic surgeon, who may precribe a course of silicone sheeting.