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

Abstract

Let $G=(V(G), E(G))$ be a graph and ${\mathcal {G}}_i$ be a class of graphs for each $i\in [k]$. A $({\mathcal {G}}_1,\ldots , {\mathcal {G}}_k)$-partition of G is a partition of V(G) into k sets $V_1,\ldots , V_k$ such that, for each $j\in [k]$, the graph $G[V_j]$ induced by $V_j$ is a graph in ${\mathcal {G}}_j$. In this paper, we prove that every planar graph without 4-cycles and 6-cycles admits an $({\mathcal {F}}_2, {\mathcal {F}})$-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.