Decomposing a graph into pseudoforests with one having bounded degree

Abstract

The maximum average degree of a graph G, denoted by $\mathrm{mad}(G)$, is defined as $\mathrm{mad}(G)={\max }_{H\subseteq G}\frac{2e(H)}{v(H)}$. Suppose that σ is an orientation of G, $G^{\sigma }$ denotes the oriented graph. It is well-known that for any graph G, there exists an orientation σ such that ${\mathrm{\Delta }}^{+}\left(G^{\sigma }\right)\le k$ if and only if $\mathrm{mad}(G)\le 2k$.

A graph is called a pseudoforest if it contains at most one cycle in each component, is d-bounded if it has maximum degree at most d. In this paper, it is proven that, for any non-negative integers k and d, if G is a graph with $\mathrm{mad}(G)\le 2k+\frac{2d}{k+d+1}$, then G decomposes into $k+1$ pseudoforests with one being d-bounded. This result in some sense is analogous to the Nine Dragon Tree (NDT) Conjecture, which is a refinement of the famous Nash–Williams Theorem that characterizes the decomposition of a graph into forests. A class of examples is also presented to show the sharpness of our result.