Which requires more energy: walking 1 km or cycling 1 km at the same speed?

This is one of those cases where the physics definition of "work done" does not match your experience.

If you move an object that is subject to gravity along a horizontal surface, physics tells us the only work done is the work done to accelerate the object, and overcome the force of friction.

Biologically, "walking" is a complex action that involves many muscles contracting and stretching. But muscles are not a reversible system (unlike a spring) - work done contracting is not returned when they stretch again (incidentally this is why some animals like kangaroos have highly elastic tendons... this greatly improves the efficiency of their jumping). So if you do a deep squat, returning to the same position, you will have expended (chemical) energy, even though you "did no net work" in the physics sense. Walking involves continuous (small) changes in the height of your center of mass, and so a lot of "shallow squats". Even if you could walk smoothly without bouncing your center of mass up and down, your legs will bend and stretch as you absorb the shocks of the road (your leg has to be straight when it is placed in front of you, and bend when it is directly underneath you - or you have to move your center of gravity).

When you look at cycling, you don't have to carry your entire weight on your legs - the only work you need to do is work required to overcome the small rolling friction (at walking speed) and air drag (if you go a little faster). Also, your center of mass stays at a constant height, so there is no energy lost in "bouncing" so much.

If you look at calories burned, this is confirmed. Riding a fast mile on a bicycle you will burn about 50 kcal, and much less if you go more slowly; running a mile will burn about 130 kcal (depending on how heavy you are - this is for 170 pound runner).

UPDATE

In Umberger et al, "A model of human muscle energy expenditure", Computer Methods in Biomechanics and Biomedical Engineering, 2003 Vol. 6 (2), pp. 99–111 the authors give a detailed model of energy expenditure of different muscles, showing clearly that load, not just extension, play a big role. And obviously when you carry your entire weight, you are carrying more of the load. They include the following diagram showing what muscles are loaded during what part of the walking cycle:

It occurred to me that the earliest "bicycle" that I am aware of was the velocipede, a contraption that allowed one to "walk" while part of the body weight was carried by the "bike". This immediately reduced the effort required for locomotion and provides further evidence for the above.

Image from that article:

Since you can travel the distance faster on a bicycle for the same amount or less of physical tiredness within the time, it's clear that the bicycle uses less energy.

I run, have been jogging and running most of my 70 years of age.Have ran many marathons and will run again in Los Angeles marathon on March 2017.

As I keep track of my own efficiency and pace over the years I am a good candidate to have personal information on this.

Running is not conserving energy like a spring or a machine made up of joints and levers that would look like a humanoid robot.

Efficiency plays a great role; age and endurance do as well. Experienced runners take advantage of the stride and bounce over the landing foot as much as possible, but even that is not going to return much of work you used to land the foot. Every movement is consuming energy and creating heat and undesired toxins in your body. Muscles which seem not contributing to your locomotion are working hard to keep your head up and your torso erect!

My time on marathon was 3:45 hrs:mins now I am happy with 5:50.
Because my muscles and joints are not as efficient as before. While I burn more energy for the same track.

When you ride a bicycle you are taking advantage of efficiency of the machine. The more efficient the machine the less energy you burn.

As a good contrast if you ride a bicycle downhill you don't use energy but if you run downhill you still burn energy and can also hurt your joints!