I think we need to get right back to basics,,,,firstly L.E.D.'s are current devices, you allow a current to flow through them and they develop their own voltage drop which is fairly independent of the current flow.
The voltage drop they develop is determined by the type of semiconducting material from which they are made, but a rough rule of thumb is...the shorter the wavelength they emit the greater the voltage drop, I think this comes from Plank's Equation, the shorter the wavelength the greater the energy per quanta, so the bluer the L.E.D. the greater the voltage drop. Bear in mind that white and warm white L.E.D.s are actually blue, violet or U.V. L.E.D.s with a phosphor in front of the die...this converts some of the short blue or violet wavelengths to longer green, yellow orange and red wavelengths which mix with the bluer ones emitted directly by the die to make a shade of white. Most L.E.D. s and L.E.D. arrays now made for general lighting work line this. (The phosphor is a yttrium aluminium garnet activated with europium I think.) But I digress...back to the problem at hand, dimming a L.E.D., or array of L.E.D.s off a Lithium Ion battery as the power source.
What you really need is a current source because L.E.D.s are current devices. The brightness of the emitted light is proportional to the current flowing through the L.E.D., so if you adjust the current, you also adjust the brightness. If you reduce the current to zero, the brightness drops to zero too.
If you wish to dim all the different coloured L.E.D.s together with one control, you will need one type of circuit. If you wish to dim the different colours separately, you will need a circuit for each and a control for each.
I will explain how to dim just ONE L.E.D. regardless of its colour, but there will be limitations. There are two ways of going about it, one has been mentioned already, P.W.M., and the other is an older technique, LINEAR! Effectively P.W.M.is a high efficiency digital technique and its core function revolves around MOSFETS or other types of transistors switching on and off rapidly. LINEAR is much less efficient, so it is far more "hungry" on the battery capacity and the controlling transistor can get very hot....but it has one HUGE advantage though...it is easy to construct for a beginner!!
Now the low battery voltage limitation (what is it?, 3.7) this makes design difficult because there is little "headroom" to move about. It is like walking around in a room 37cm high, you can't place anything higher than that...least not easily anyway.
So depending on what you want these dimmable L.E.D.s are for depends upon just how much effort you will expend to achieve it. If it is some sort of demonstration, only ever to be used once or twice to prove a point, principle or degree of acquired understanding...then you can cut quite a few corners in the design and make the whole lot easier. BUT, if these dimmable L.E.D.s are some mission-critical application and the city's transport network will grind to a halt without them, then a significant amount of effort would be expended upon the design to realize something of great reliability and ability to function for decades in a harsh environment.
I will assume the former, a demonstration of the ability to dim L.E.D.s of moderate power just a few times using an inefficient linear technique.
So, in this case you simply need a constant current source and a pass transistor....many bench D.C. power supplies come with this feature as standard.
Now you cannot simply use a potentiometer in series with the L.E.D., it simply does not possess the power rating to handle the current. Electronics will be required, and that means at least one active device...that is a MOSFET or transistor than can be controlled by the pot, but it does all the "heavy lifting".
But you still have that issue of very low voltage headroom....and it is definitely beyond your scope to engineer a circuit to perform this function with such low headroom.....so to get more headroom, you MUST place two batteries in series to get 7.2 volts odd....you now have the headroom to design the rest.
You will need a circuit called a "voltage controlled current source" it consists of an op-amp, a transistor and a resistor...I will describe it, but there are loads of catches, the transistor is a simple P.N.P. bipolar transistor its emitter goes up via the resistor to the 7.2v rail. Its base connects to the output of the op-amp and the non-inverting (+) input of the op amp is connected back to the emitter....the collector is the controlled current output and the inverting (-) input of the op-amp is the voltage control input. (And that is the basic V to I converter, but you will need some extras to make it practical.)
Now in your case this circuit alone won't deliver enough current to do what you want so you will need a BIG CHUNKY pass transistor....an N.P.N. and the current gain of this will allow the L.E.D. to be driven...so the collector of the small P.N.P. will go into the base of the big N.P.N., the L.E.D.s anode will connect to the emitter of the big N.P.N. and its cathode to the 0v rail or negative pole of the battery. In the collector of the big N.P.N. put a low value resistor, 0.5 of an ohm or so and then up to the 7.2v. Get a fairly high value pot, 10K say, connect one end of it to the 7.2 volts, connect its wiper to the inverting input of the op-amp via a 10k resistor. Connect the other end of the pot to 0v via a resistor...start with 10K too...but you might change it later. Make the resistor at the emitter of the P.N.P. about 56 ohms.
Put a BIG heatsink on the big NPN transistor...it will get hot! Use something like an old 2N3055. For the PNP, use something like a 2N2904 or BC556/7/8. The op-amp has to be fairly special, it has to have the ability to have the output swing almost rail to rail and it has to be a singe rail type...similar to one quarter of an LM324, but this is possibly not adequate.
Now it will probably not work first off...it'll do something daft like change brightness suddenly right up one end of the pot's rotation...if so increase that 10K between the bottom end of the pot and 0v to 100K. If things still don't go as you expect them, keep tweaking....put a 560 ohm resistor at the top end of the pot. (between it and the 7.2v) and see how that effects it...you have to have loads of patience, possibly destroy a few components...but that will teach you their limitations...and by fiddling and adjusting things and see what happens, you will get a feel for what is happening. Read up on op-amps, read up on current sources, read up on voltage controlled current sources of this type I have described and you will learn something but it will be slow....don't expect to "fly to the moon tomorrow".
If you get the hang of the linear dimmer, then you might be ready to go for the better, more efficient P.W.M. option using a 7555 driving a Logic Level MOSFET possibly with a single capacitor-inductor low pass filter to get rid of the strobing of the L.E.D. inherent with this type of digital power control.
What is important is to get something back for all your effort, it may not work perfectly as you expected it to, but you can make small changes in the component values and see how that changes things...if you've got a bit, then you can get a bit more by taking small steps...you will learn which way to step and which steps lead to the result you desire or at least nearer to it and which lead way to the point of circuit disfunction and ultimate disappointment! The hardest bit is actually making that first step from nothing at all to something, even if it is far from perfect.
I found the same circuit here
http://schematics.dapj.com/2009/02/current-source-for-resistance.html but different component values and being used for a different purpose.
