On the List Coloring Version of Reed's Conjecture

Abstract

The chromatic number of a graph is bounded below by its clique number and from above by its maximum degree plus one. In 1998, Reed conjectured that the chromatic number is at most halfway in between these trivial lower and upper bounds. Moreover, Reed proved that its at most some non-trivial convex combination of the two bounds. In 2012, King and Reed produced a short proof that, provided the maximum degree is large enough, a combination of 1/130,000 suffices. Recently Bonamy, Perrett, and the second author improved this to 1/26.It is natural to wonder if similar results hold for the list chromatic number. Unfortunately, previous techniques for ordinary coloring do not extend to list coloring. In this paper, we overcome these hurdles by introducing several new ideas. Our main result is that the list chromatic number is at most some non-trivial convex combination of the clique number and the maximum degree plus one. Furthermore, we show that for large enough maximum degree, that a combination of 1/13 suffices. Note that this also improves on the best known results for ordinary coloring.