We prove a conjecture of Ohba that says that every graph G

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)||V(H)|−1 over all subgraphs H with at least two vertices. Generalizing the Nash‐Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb (G)≤k+dk+d+1, then G decomposes into k+1 forests with one having maximum degree at most d. The conjecture was previously proved for (k,d)∈{(1,1),(1,2)}; we prove it for d=k+1 and when k=1 and d≤6. For (k,d)=(1,2), we can further restrict one forest to have at most two edges in each component. For general (k,d), we prove weaker conclusions. If d>k, then Arb (G)≤k+dk+d+1 implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If d≤k, then Arb (G)≤k+d2k+2 implies that G decomposes into k+1 forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.

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A parity walk in an

The dichromatic number of a graph G is the maximum integer k such that there exists an orientation of the edges of G such that for every partition of the vertices into fewer than k parts, at least one of the parts must contain a directed cycle under this orientation. In 1979, Erdős and NeumannLara conjectured that if the dichromatic number of a graph is bounded, so is its chromatic number. We make the first significant progress on this conjecture by proving a fractional version of the conjecture. While our result uses a stronger assumption about the fractional chromatic number, it also gives a much stronger conclusion: if the fractional chromatic number of a graph is at least t, then the fractional version of the dichromatic number of the graph is at least 1 4 t/ log 2 (2et 2 ). This bound is best possible up to a small constant factor. Several related results of independent interest are given.