关于图分解的Stiebitz型结果的推广
A generalization of Stiebitz-type results on graph decomposition
摘要：
本文研究了最小度约束下重图的分解问题，给出了几个结果的统一推广。设$G$是一个多重图，其中没有四边形与三角形和其他四边形共享边，设$μ_G(v)=\max\{μ_G(u,v):u\in V(G)\setminus\{v\}\}$，其中$μ_G(u,v)$是$G$中连接$u$和$v$的边数.证明了对于任意两个函数$a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$，如果每个$v\in V(G)$有$d_G(v)\ge a(v)+b(v)+2μ_G(v)-3$，则存在$V(G)$的一个分区$(X,Y)$，使得每个$x\in X$有$d_X(x)\geq a(x)$，每个$y\in Y$有$d_Y(y)\geq b(y)$.这将Diwan [3]，Liu和Xu [7]，Ma和Yang [10]关于简单图的相关结果推广到了多重图的情形.
英文摘要：
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 $μ_G(v)=\max\{μ_G(u,v):u\in V(G)\setminus\{v\}\}$, where $μ_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\rightarrow\mathbb{N}\setminus\{0,1\}$, if $d_G(v)\ge a(v)+b(v)+2μ_G(v)-3$ for each $v\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such that $d_X(x)\geq a(x)$ for each $x\in X$ and $d_Y(y)\geq b(y)$ 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.
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