Planar Graphs with Sparse Triangles and Without 4-Cycles and 5-Cycles Admit (\ (F₂, F₇\) ) -Partition

If the vertex set of a graph G can be partitioned into k subsets V1,V2,…,Vk, and the induced subgraph on each subset Vi is a forest whose maximum degree is at most di (i=1,…,k), then this partition is called an (Fd1,…,Fdk)-partition of G. Cho et al. (2021) conjectured that every planar graph without 4-cycles and 5-cycles admits an (F2,Fd)-partition, where d is a positive integer. In this paper, we prove that every planar graph with sparse triangles and without 4-cycles and 5-cycles admits (F2,F7)-partition. This result provides further support for the above conjecture.