Partition of Sparse Graphs into Two Forests with Bounded Degree

Abstract

Borodin and Kostochka proved that for $d_2 \geq 2d_1+2$ and a graph $G$ where every subgraph $H$ satisfies $$ e(H) < \left(2 - \frac{d_2+2}{(d_1+2)(d_2+1)}\right)n(H) + \frac{1}{d_2+1} $$ has a vertex partition $V(G) = V_1 \cup V_2$ such that $G[V_i]$ has maximum degree at most $d_i$ for each $i$. We show that under the same conditions we can additionally conclude that each $G[V_i]$ is a forest.

Theorems & Definitions (40)

• Theorem 1.1: Borodin and Kostochka BorodinKostochka2014
• Theorem 1.2
• Corollary 1.3
• Theorem 2.1
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3
• Corollary 3.4