Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree

Abstract

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.

Visibility Number of Directed Graphs

“…They proved this for k = 1 when d ∈ {1, 2}. In [16], it was proved also for d = k + 1 and for k = 1 when d ≤ 6.…”

Degree Bounded Forest Covering

“…Partial results towards this weaker conjecture are obtained in [7,1,6]. Recently, for ǫ ≥ 1 2 , Conjecture 2 was shown to be true by Kim et al [4], but the case ǫ < 1 2 remains open (there are some special values for which it is known, see [4]). Our main result is the following theorem which almost proves Conjecture 2.…”

“…However, this is less interesting because in this case Conjecture 2 itself has been proved in [4], so we only sketch the proof at the end of Section 3.…”

“…Partial results are obtained by combinatorial method [6,4] and by topological method [3], and related results are known for planar graphs [1,2]. In this paper we are interested in a weaker form of the conjecture, where the bounded degree subgraph is not required to be a forest.…”

Decomposing a graph into pseudoforests with one having bounded degree

“…For the progress of Nine Dragon Tree (NDT) Conjecture, in [7], Montassier et al proved the cases of (k, d) * = (1, 1) and (k, d) * = (1, 2), and they showed that no larger value of γ f (G) is sufficient; in [6], Kim et al proved the cases of (k, d) * = (k, k + 1) and…”