Partitions of multigraphs under degree constraints

Abstract

In 1996, Michael Stiebitz proved that if $G$ is a simple graph with $\delta(G)\geq s+t+1$ and $s,t\in \mathbb{Z}_{\geq 0}$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $\delta(G[A])\geq s$ and $\delta(G[B])\geq t$. In 2016, Amir Ban proved a similar result for weighted graphs. Let $G$ be a simple graph with at least two vertices, let $w:E(G) \to \mathbb{r}_{>0}$ be a weight function, let $s,t \in \mathbb{R}_{\geq 0}$, and let $W=\max_{e\in E(G)} w(e)$. If $\delta(G)\geq s+t+2W$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $\delta(G[A])\geq s$ and $\delta(G[B])\geq t$. This motivated us to consider this partition problem for multigraphs, or equivalently for weighted graphs $(G,w)$ with $w:E(G) \to \mathbb{Z}_{\geq 1}$. We prove that if $s,t\in \mathbb{z}_{\geq 0}$ and $\delta(G)\geq s+t+2W-1\geq 1$, then $V(G)$ can be partitioned into two sets $A$ and $B$ such that $\delta(G[A])\geq s$ and $\delta(G[B])\geq t$. We also prove a variable version of this result and show that for $K_4^-$-free graphs, the bound on the minimum degree can be decreased.