Defective 2-colorability threshold for planar graphs excluding 3-, 4-, and 6-cycles

Determine whether every planar graph without 3-, 4-, and 6-cycles is (d1, d2)-colorable whenever d1 + d2 ≥ 6, thereby establishing the proposed defective-coloring threshold in full.

Background

The paper studies defective 2-colorings of planar graphs that contain no cycles of lengths 3, 4, or 6. In a (d1, d2)-coloring, vertices assigned color i have at most di neighbors assigned the same color. Prior results established that these graphs are (0,6)-colorable and (3,3)-colorable, while the paper proves the additional case (2,4)-colorability.

The conjecture asserts that the condition d1 + d2 ≥ 6 is sufficient for (d1, d2)-colorability for every planar graph excluding 3-, 4-, and 6-cycles. Theorem 1.2 verifies only the specific parameter pair (2,4), so the conjecture remains only partially established in the paper.