Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-01-13 , DOI: 10.1016/j.disc.2019.111797
Huajing Lu; Xuding Zhu

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

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