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  "body": "\n## Statement\n\nIn 1964, Nash-Williams generalized the above result to forests:\n\n\u003e A graph $G$ can be partitioned into $t$ edge-disjoint forests iff for every $U \\subset V(G)$, the [[induced subgraph|0EREPBVG2G]] $G[U]$ has at most $t(|U|-1)$ edges.\n\nOther proofs are given here.\n\nThis is how people usually define what it means for a graph to be ***t*-arboric.**\n\nIn other words, for every subgraph $S=G[U]$, we have $t \\geq \\lceil E(S) / (V(S) - 1) \\rceil$. It is tight in that there is a subgraph $S$ that saturates the inequality (or else we can choose a smaller $t$). This leads to the following formula\n\n\u003e $$t = \\lceil \\max_{S \\subset G} \\frac{E(S)}{V(S) - 1} \\rceil,$$\n\nalso referred to as the **Nash-Williams formula.**\n\nThe general problem is to ask when a graph can be covered by edge-disjoint subgraphs.\n",
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