Foreword

Summary

It has been more than thirty years since I first encountered the Circuit Double Cover Conjecture. A colleague had approached me in the hallway, with what he then referred to as “a nice little problem.” Indeed, even back then, the problem had already been floating here and there for quite some time. Communication, however, was not remotely what it is today and new ideas spread around erratically and at a very slow pace.

It did not seem difficult. I thought, at first, that he meant it to be an exercise for our Graph Theory course, and was somewhat embarrassed, as I was not able to solve it right away. “Every edge doubled,” I was thinking out loud, “that makes an Eulerian graph. … bridgeless, so there should be a simple way to construct a circuit partition, that avoids both copies of an edge on the same circuit… Well, I will think about it.”

Three decades and hundreds of related publications later, and I still think about it, and so do many others. Indeed, a fascinating “nice little problem.”

No serious mathematical question, solved or unsolved, is as simple to state and as easy to understand. No background is required; nothing essential about graphs; not even basic arithmetic. Take the intuitive concept of a line joining two points, the idea of following such lines back to the starting point to form a circuit, the ability to count “one, two” and voilà, you have the Circuit Double Cover Conjecture.