A generalization of Stiebitz-type results on graph decomposition

Abstract

In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let ${\mu }_G(v)=\max \{{\mu }_G(u,v):u\in V(G)\setminus \{v\}\}$, where ${\mu }_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\to \mathbb{N}\setminus \{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2{\mu }_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(x)\ge a(x)$ for each $x\in X$ and $d_Y(y)\ge b(y)$ for each $y\in Y$. This extends the related results due to Diwan [3], Liu and Xu [7] and Ma and Yang [10] on simple graphs to the multigraph setting.