This paper proves that if G is a planar graph without 4-cycles and l-cycles for some l ∈ {5, 6, 7}, then there exists a matching M such that AT (G − M ) ≤ 3. This implies that every planar graph without 4-cycles and l-cycles for some l ∈ {5, 6, 7} is 1-defective 3-paintable.

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A short cycle means a cycle of length at most 7. In this paper, we prove that planar graphs without adjacent short cycles are 3-colorable. This improves a result of Borodin et al. (2005).