every planar graph can be expressed as a union of (maximum) five forests Can you help me prove that every planar graph can be expressed as a union of (maximum) five forests?  Ι used induction on the number of vertices. A planar graph $G$ has at most $3|V|-6$ edges, so there is at least one vertex $v$ of degree 5 or less.  Let $\{w_1,\ldots, w_j\}$; $j \le 5$ be $v$'s neighbors in $G$.  Then let $F_1,\ldots, F_5$ be the 5 edge-disjoint forests (some may be empty) such that $\cup_{j=1}^5 F_i = E(G \setminus \{v\})$. How do I prove that $F_1,\ldots, F_5$ exist?  Then for each $i \le j$ set $F'_i = F_i + \{v,w_i\}$, and (if $j$ is strictly less than 5) for each $i \in \{j+1,\ldots, 5\}$ set $F'_i=F_i$. Then each $F'_i$ is also a forest, and that each edge of $G$ is in one of the $F'_i$s.  Special thanks to user Mike for the answer.
every planar graph can be expressed as a union of five edge-disjoint forests How can I prove that every planar graph can be expressed as a union of (maximum) five edge-disjoint forests?  Only thing I can think about is going somehow like the $5$ color theorem but I get stuck in the middle. Or with induction, but still something doesn't fit. Can you help?
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