Would you run faster on Mars? Given the 1/3rd of Earth's gravity on Mars and neglecting space suit limitations and also assuming you have maintained your muscle strength, would you run faster on Mars?
The answer may not be so straightforward.
This is similar to the reduced 1/6th gravity on moon compared to the 1/3rd on Mars. We all have seen the video footage from the moon landings and walks on the moon surface. The astronauts appear to have increased foot strength but they have in a rather slow general movement, especially evident when they are waving their hands. This is maybe because absent or reduced gravity (microgravity) makes you actually float like inside water. Your limbs' muscles are constantly fighting your own mass's inertia
(buoyancy replaced by the word inertia) and your feet maybe stay longer on "air" not touching the ground since you are not assisted by the Earth's gravity increased downward force. So I guess this is an open question.

image source: https://www.pinterest.com/pin/763078730595604862/
Update 7 May 2022: Seems this question has risen quite a debate in the last couple of days. I did some more digging in the literature and could find only one directly dedicated publication to this question about running on Mars:
https://pubmed.ncbi.nlm.nih.gov/15856558/ (Abstract only)
Also about running on the moon this research here says that experiments have shown that the maximum speeds achieved will be much greater than initially theoretical predicted mainly due to the extra momentum gained by the hands movement, but still inferior speeds to Earth's gravity:
https://www.theverge.com/2014/9/17/6353517/nasa-astronauts-tested-how-fast-humans-can-run-on-the-moon
 A: You question is interesting but perhaps a bit ill-posed.  Consider, for example, the difference between a 100-m sprinter and a miler.  The sprinter will lean as far forwards as possible so as to accelerate as fast as possible, while the miler reaches operational speed early in the race.
If I were on Mars, I'd use spikes or whatever to get a good grip on the ground, and lean forward at a large angle so that all but $\epsilon$ of my applied force vector is in the horizontal direction.  The lack of atmosphere lets me run with less resistance;  the difference in gravitational force is cancelled out by my attack angle.
A: Summary
All else being equal, runners on Earth will accelerate harder up front, but runners on Mars will have higher top speeds. As such, longer runs will favor Mars, while shorter runs will favor Earth. Assuming you can breathe on Mars, or you're the Terminator and don't need to.
I suspect that initial acceleration will be more important, and Earth runners will win more, in races short enough to typically be considered "sprinting".
Basic Physics
There are three effects in play here before you get into human physiology.
First, Mars has no oxygen to breathe, so you'd have trouble running at all. Let's assume the person has a magical oxygen supply of negligible mass. We could also be talking about a battery-powered electric vehicle, android, etc.
Second, Mars has very little atmosphere, so you're not limited by air resistance nearly as much as on Earth. However, in dirt or gravel, rolling resistance can be a lot higher than air resistance, so this may not matter.
Third, Mars has less gravity, giving less traction, resulting in a lower maximum acceleration force. On a loose surface, you can dig in and push against the ground, but ultimately low gravity will make the dirt stick to the surface less, and still give you worse acceleration.
Compounding Factors
Going uphill means traction is more important. Gravity differences will tend to dominate, and Earth vehicles accelerate harder. Because there's now a rearward acceleration force (the vehicle's weight), maximum speed will be limited by the speed at which the acceleration force equals weight, which will be the dominating limit for humans or low-powered vehicles, or steep inclines.
Going downhill means traction is less important. There will be a point on the downhill where power is essentially irrelevant and traction just doesn't matter. At that point, lower gravity will accelerate at a lower rate, but lower air density on Mars means terminal velocity will be much higher.
On hard, flat surfaces, traction will be the main factor in initial acceleration, while air resistance will be the main factor in top speed.
In deep sand or other loose terrain, rolling resistance will dominate air resistance. On flat ground or uphill, Mars' lower gravity won't sink you into the ground as far, and you'll tend to do better there. There will be a particular downhill slope where the gravity pulling you forward is more important than that sinking you into the dirt, and you'll do better on Earth.
Running
A wheeled vehicle is different than feet moving across the ground, but the differences aren't really significant here. Feet will tend to dig into dirt better, but so will mudding tires. Tires are more efficient, but relative efficiencies are still worse in the same places. So we can treat running in much the same way we'd expect a low-powered vehicle to perform, as far as the foot-to-ground interface is concerned.
However, there is a major difference between humans and the vehicles we build: gearing. A bike or car can be re-geared for better acceleration or top speed as needed. A bike on Mars can have multiple gears to keep the rider near their maximum power at all speeds, resulting in both maximum acceleration and top speed (or close to it). A car can do the same.
But a human only has one "gear". Regardless of potential power output, there is a maximum speed we can twitch our legs back and forth. The linear speed of our feet at that frequency is the outside maximum we can run. For most humans, the issue is diminishing power output at higher frequencies, to the point where all the power expended is used to maintain that leg speed, and there's nothing left for acceleration. For Olympic-level sprinters, the speed of nerve impulses starts making it impossible to think about moving faster.
This means that even on a downhill stretch, maximum speed is limited. If we try to exceed that speed, we'll just tumble and hurt ourselves. It also means we very quickly get to a speed where we can't overcome traction, even in low gravity. As such, a lot of the issues above become irrelevant.
So on a downhill, we'll accelerate harder on Earth (better initial grip plus gravity accelerates us directly). Because Mars has lower gravity, less power is used fighting gravity and more is therefore available for moving our legs, so we'll go a little faster top speed. But probably not by much.
On an uphill, Mars will win at acceleration, because we're fighting gravity less. Earth might give better instantaneous acceleration because of the better grip, but after a step or two Mars will be in the lead. Mars will also win at top speed since less power is wasted fighting gravity.
On flat ground, Earth will win off the line and hold the lead on speed for a while. Both places will have similar top speeds, so Earth will tend to win because of the initial distance gains. However, Mars still requires less wasted energy fighting gravity, so top speed will again be higher.
Conclusion
A runner on Earth will get a better acceleration off the line, but lower top speed. A Mars runner will win if the race is long enough. The more downhill the race, the longer it will take before the top speed advantage allows the Mars runner to overcome Earth's acceleration advantage.
A Note on Vertical Force
A number of sites around the internet seem to think runners go faster through vertical force. This is not really true. Because gravity is pulling the runners down, some vertical force is required to not fall over, but it's impossible to move forward only using vertical force.
Instead, runners are hitting the ground diagonally. Some of the force is used to propel them forward, and some is used to keep them upright. At low speeds, there's more power available (less is wasted moving their legs), and the vertical power requirement is constant, so more power is available for forward acceleration. At higher speeds, more power is required to keep their legs moving, so a greater portion of the power remaining is required to keep them upright. At top speed, they're just maintaining speed, so there's only a small amount of forward power being used to overcome air resistance.
We can see this in the angle of the runner's bodies. Off the line, runners are leaned way forward to balance against the intense acceleration. At top speed, runners are nearly vertical because there's no acceleration. But top speed is a result of how fast the runners can cycle their legs while staying upright, which is a result of how much power is left over after vertical power is taken into account.
If nerve cycling limits are in effect, the guy with longer legs wins, because each stride carries him farther and power stops being the deciding factor at all.
And you can't go faster by just jumping higher. Jumping higher means more time letting air resistance slow you down. The fastest sprinters will be airborne just long enough for the next step to be ready. Airborne less time means you trip and fall (or waste energy preventing said trip and fall). Airborne more time means more time you're not actively pushing forward to maintain speed.
On Mars, you'll waste less power staying airborne, so you'll have more power to propel yourself forward. But you'll still spend a bunch of power just cycling your legs at top speed, so the slight difference in air resistance won't make too much difference.
A Note on Human Legs
The above assumes runners on Mars would use the same motion as running on Earth. However, Mars' lower gravity means it will be optimal to lean forward more, which means the leg may not be at its optimal angle for transfer of power. I don't pretend to know how this affects things, but I suspect the answer is "not much". Once at top speed, the runner will be mostly vertical in either case.
References
This article, referenced in another answer by Anders Sandberg, shows experimentally what I've suggested above. Runners in low gravity automatically reduce the vertical force used so they bounce as minimally as possible. This is another way of saying they'll stay airborne just long enough for the next stride to be ready.
A: This is a partial answer.
Anyone who has run on an Alter-G treadmill will tell you that it is much easier to run at a given speed when your weight is artificially reduced in the way that Alter-G's do. I am sure they are not a perfect simulation of lower gravity, but it's a sanity check on the question.
As has been mentioned, an important distinction is the kind of running you are doing.

*

*Sprinting (such as the 100m dash) is essentially limited by the power you are able to apply to the ground, and air resistance. I can't speak to how sprinting would be affected in lower gravity.


*Endurance running (such as a marathon, or generally everyday moderate paced running) is largely limited by energy expenditure. Although, there are many other relevant factors.
In endurance running, energy expenditure is largely measured by volume of oxygen consumed per minute. This also correlates with heart rate and perceived effort.
This study and many that it cite show that on "antigravity treadmills" such as the AlterG, decreasing the apparent bodyweight leads to less oxygen consumption. In other words, this suggests that for endurance running, all else equal, the same effort level would lead to a faster pace in low gravity.
(Decreased air resistance on Mars would also increase your speed, as long as you are still somehow breathing an Earth mixture of air.)
Edit: I would argue the decreased energy expenditure makes sense from first principles, because moving horizontally on a flat surface at a constant speed requires no work regardless of gravity. So the question is whether increased gravity causes the bounding motion of running to waste more or less energy, and it would be very surprising if it were less.
A: The speed of walking and running depend on pendulum-like motion of the legs. If you walk at different speeds the power used varies, and has a minimum roughly corresponding to the free pendulum motion of your legs.
That swing time is $T\approx 2\pi \sqrt{L/g}$, and since each step has a length proportional to your leg length $L$ the speed scales as $v\propto L/T = \sqrt{gL}$. So you will tend to walk more slowly in low gravity. This is complicated by people talking longer steps in lower gravity. This can be tested using parabolic flight or having weight-reducing spring suspensions.
However, running involves moments when both legs are in the air. It becomes energetically favourable for bipeds when the Froude number is $\approx 0.5$, or $v=\sqrt{g L/2}$. So at lower gravity you start running at lower speeds, which also checks out with suspended runners. The energetically most favourable speed is at Froude = 1/4, which means you will tend to run at a speed scaling as $\sqrt{g}$. So Martian runners would tend to run at about 60% of Earth runner speeds.
Low gravity running also involves a flatter trajectory with less bouncing, and has a reduced energy cost compared to walking: on Mars people may be running more, but do it more slowly.
A: Buoyancy is not what slows the motion of astronauts on the Moon. Buoyancy requires some heavier-than-you liquid to surround you (in order to impede you significantly). When vacuum surrounds you there's no buoyancy.
They aren't floating. In fact they don't face air drag. They move slow, because the space suits impede them. Also they had to be very careful, e.g. not to trip and fall, damaging the space suit.
If the space suit wasn't an issue, you can move in large jumps, making your movement faster. Low atmospheric pressure on Mars also means there's no air drag, which also means you can move pretty fast.
A: (a fork of Carl’s answer)
Your question is interesting but perhaps a bit ill-posed. Consider, for example, the difference between a 100-m sprinter and a marathoner or jogger. The sprinter will lean as far forwards as possible so as to accelerate as fast as possible, while the marathoner reaches operational speed early in the race.  So kind of two questions.; I'll focus on the sprinter.
If I were sprinting on Mars, I'd use spikes or whatever to get a good grip on the ground, and lean and run BACKward at a large angle so that all but 
ϵ
of my applied force vector is in the horizontal direction. The lack of atmosphere lets me run with less resistance; the difference in gravitational force is cancelled out by my attack angle, which will be higher than Carl’s because my knees won’t be in the way.
Consider: To achieve maximum acceleration and speed, I expect it'll be essential to maximize the portion of the force exerted against the ground that is in the horizontal direction, which is done by leaning and pushing mostly horizontally.  The force's down component can push the sprinter off the ground; if it's too large, the sprinter has to wait to come back down into ground contact before taking another step.  Also, this allows a larger component of the force to increase/maintain speed, so maximum speed should be greater.
Breakdown under Earth gravity, time for each 10m of record 100m dash:

lack of gears does seem likely to dominate.
