Partitioning Planar Graphs Without Specific Cycles into a Forest and a Disjoint Union of Paths

In this paper, we show that if a planar graph G satisfies the following conditions: (i) none of its 3-faces is adjacent to a 6−-face, and (ii) none of its 4-faces is adjacent to a 5−-face, then V(G) can be partitioned into two subsets, each induces a forest, while one of the forests has maximum degree of at most 2. This result implies that every planar graph (i) with no chordal 7−-cycles, or (ii) with neither 4- nor 6-cycles, also admits such a partition.