As they say, "The basic idea of the proof is the same as Appel and
Haken's." They break the theorem down into two parts, such that "The first
proof needs a computer. The second can be checked by hand in a few months, or,
using a computer, it can be verified in about 20 minutes."
They provide URLs for further details, plus for the program itself,
and for an independent set of programs for the same algorithm, written by
Gasper Fijavz under the guidance of Bojan Mohar.