Partitioning Planar Graphs without 4-Cycles and 6-Cycles into a Linear Forest and a Forest

Let G = (V(G), E(G)) be a graph and G(i) be a class of graphs for each i is an element of [k]. A (G(1), . . . , G(k))-partition of G is a partition of V (G) into k sets V-1, ... , V-k such that, for each j is an element of [k], the graph G[V-j] induced by V-j is a graph in G(j). In this paper, we prove that every planar graph without 4-cycles and 6-cycles admits an (F-2, F)-partition. As a corollary, V (G) can be partitioned into two sets V-1 and V-2 such that V-1 induces a linear forest and V-2 induces a forest if G is a planar graph without 4-cycles and 6-cycles.