# colorings – What is a guarenteed amount of colors, depending on the graph’s arboricity

And, you can do better than $$f(a) = 2a$$. For example, take a complete graph on $$4$$ vertices: $$a,b,c,d$$. The arboricity is $$2$$ since $$(a,b), (b,c),$$ and $$(c,d)$$ forms the first tree, and the remaining edges forms a second tree. The graph is colorable with exactly four colors.