Minimal counterexamples and discharging method

Recently, the author found that there is a common mistake in some papers by using minimal counterexample and discharging method. We first discuss how the mistake is generated, and give a method to fix the mistake. As an illustration, we consider total coloring of planar or toroidal graphs, and show that: if G is a planar or toroidal graph with maximum degree at most κ − 1, where κ ≥ 11, then the total chromatic number is at most κ.