Partition of Sparse Multigraphs into a Forest and a Forest with Restrictions

TL;DR

This paper investigates how to partition sparse multigraphs into two forests with additional restrictions, providing exact bounds on parameters for various classes of graphs and constraints.

Contribution

It establishes precise bounds on (a,b) parameters ensuring such partitions with restrictions, extending known results to new classes and constraints.

Findings

Exact bounds for (a,b) ensuring partitions into forests with degree constraints.

Results applicable to both multigraphs and simple graphs for D ≥ 2.

Extension of known sparsity partition results with new restrictions.

Abstract

The following measure of sparsity of multigraphs refining the maximum average degree: For $a>0$ and an arbitrary real $b$, a multigraph $H$ is \emph{$(a,b)$-sparse} if it is loopless and for every $A\subseteq V(H)$ with $|A|\geq 2$, the induced subgraph $H[A]$ has at most $a|A|+b$ edges. Forests are exactly $(1,-1)$-sparse multigraphs. It is known that the vertex set of any $(2,-1)$-sparse multigraph can be partitioned into two parts each of which induces a forest. For a given parameter $D$ we study for which pairs $(a,b)$ every $(a,b)$-sparse multigraph $G$ admits a vertex partition $(V_1, V_2)$ of $V(G)$ such that $G[V_1]$ and $G[V_2]$ are forests, and in addition either (i) $\Delta(G[V_1])\leq D$ or (ii) every component of $G[V_1]$ has at most $D$ edges. We find exact bounds on $a$ and $b$ for both types of problems (i) and (ii). We also consider problems of type (i) in the class of simple graphs and find exact bounds for all $D\geq 2$.